Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds

*Fusun Nurcan*

#### **Abstract**

The aim of this work is to study on quasi conformally flat quasi Einstein-Weyl manifolds. In this book chapter, firstly, an interesting relationship between complementary vector field and generator of the quasi Einstein-Weyl manifold is obtained and supported by an example. Then, it is investigated that quasi conformally flat quasi Einstein-Weyl manifolds are of quasi constant curvature, recurrent and semisymmetric under which conditions after obtaining the expression of the curvature tensor of the quasi conformally flat quasi Einstein-Weyl manifold. Furthermore, some equivalences are obtained between to be of quasi constant curvature and to be semisymmetric in quasi conformally flat quasi Einstein-Weyl manifolds.

**Keywords:** quasi Einstein-Weyl manifold, Weyl manifold of quasi constant curvature, quasi conformally flat manifold, recurrent manifold, semi-symmetric manifold

#### **1. Introduction**

In 1918, H. Weyl generalized Riemannian geometry as a new way to formulate the unified field theory in physics and defined Weyl manifolds with conformal metric and symmetric connection [1]. After this study, Weyl manifolds attracted the attention of many mathematicians. In 1943, E. Cartan defined Einstein-Weyl manifolds and studied three-dimensional Einstein-Weyl spaces [2]. In 1985, P.E. Jones and K.P. Tod have studied Einstein-Weyl spaces, and then they have done many studies on this subject [3]. Although Weyl's theory did not attract much attention in physics, it attracted the attention of mathematicians and studies have been carried out on this subject until today.

An n-dimensional Weyl manifold *M* is defined as a manifold with a torsion-free connection Γ and a conformal metric tensor *gij*, if the compatible condition is in the form of

$$\nabla\_k \mathbf{g}\_{ij} - \mathbf{2g}\_{ij} \Phi\_k = \mathbf{0} \tag{1}$$

which is equivalent to

$$
\nabla\_k \mathbf{g}^{\ddagger} + \mathbf{2} \mathbf{g}^{\ddagger} \Phi\_k = \mathbf{0}, \tag{2}
$$

where Φ*<sup>k</sup>* is a complementary covariant vector field [4]. Such a Weyl manifold is denoted by *M*, *gij*, Φ*<sup>k</sup>* and (1) tells us that a Riemannian manifold is obtained if Φ*<sup>k</sup>* ¼ 0 or Φ*<sup>k</sup>* is gradient.

Φ*<sup>k</sup>* changes by

$$
\tilde{\Phi}\_k = \Phi\_k + \partial\_k(\log \lambda) \tag{3}
$$

under the transformation of the metric tensor *gij* in the form of

$$
\tilde{\mathbf{g}}\_{\vec{\text{ij}}} = \lambda^2 \mathbf{g}\_{\vec{\text{ij}}} \tag{4}
$$

where λ is a point function [4]. With reference to this transformation, the quantity *A* is called a satellite of *gij* with the weight of {*p*} if it changes by [5]

$$
\tilde{A} = \lambda^p A \tag{5}
$$

and the quantity ∇\_ *kA* is called prolonged covariant derivative of the satellite *A* of *gij* with the weight of {*p*} if it is defined by [5]

$$
\dot{\nabla}\_k A = \nabla\_k A - p\Phi\_k A. \tag{6}
$$

From (1), (4) and (6), we have

$$
\dot{\nabla}\_k \underline{\mathbf{g}}\_{ij} = \mathbf{0} \tag{7}
$$

which *gij* is with the weight of {2}.

The coefficients Γ*<sup>i</sup> jk*'s of a torsion-free connection Γ on the Weyl manifold *M*, *gij*, Φ*<sup>k</sup>* are given by

$$
\Gamma^i\_{jk} = \begin{Bmatrix} i \\ jk \end{Bmatrix} - \left( \delta^i\_j \Phi\_k + \delta^i\_k \Phi\_j - \mathcal{g}\_{jh} \mathbf{g}^{ih} \Phi\_h \right) \tag{8}
$$

where *i jk* 's are the Christoffel symbols of second kind [4].

The curvature tensor *R<sup>h</sup> ijk* of the symmetric connection Γ on the Weyl manifold is defined by

$$R^h\_{\vec{i}jk} = \partial\_{\vec{j}} \Gamma^h\_{ik} - \partial\_k \Gamma^h\_{\vec{i}\vec{j}} + \Gamma^h\_{r\vec{j}} \Gamma^r\_{ik} - \Gamma^h\_{rk} \Gamma^r\_{\vec{i}\vec{j}}.\tag{9}$$

The Ricci tensor *Rij*, which is defined by *Rij* <sup>¼</sup> *<sup>R</sup><sup>h</sup> ijh*, satisfies

$$R\_{[\vec{\eta}]} = n \nabla\_i \Phi\_{\vec{\eta}]} = \frac{1}{2} R^r\_{r\vec{\eta}}.\tag{10}$$

With the help of (9), the conformal curvature tensor *C<sup>h</sup> ijk* and the concircular curvature tensor *C*~ *<sup>h</sup> ijk* of a torsion-free connection Γ on the Weyl manifold are expressed by

*Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds DOI: http://dx.doi.org/10.5772/intechopen.105683*

$$
\begin{split}
\dot{C}^{h}\_{ijk} &= R^{h}\_{ijk} - \frac{1}{n} \delta^{h}\_{i} R^{r}\_{jik} + \frac{1}{n-2} \left(\delta^{h}\_{j} R\_{ik} - \delta^{h}\_{k} R\_{ij} + \operatorname{g}\_{ik} \delta^{hm} R\_{mj} - \operatorname{g}\_{j\bar{\mathcal{g}}} \delta^{hm} R\_{mk}\right) \\ &- \frac{1}{n(n-2)} \left(\delta^{h}\_{j} R^{r}\_{rki} - \delta^{h}\_{k} R^{r}\_{rji} + \operatorname{g}\_{ik} \delta^{hm}\_{\bar{\mathcal{g}}} R^{r}\_{rjm} - \operatorname{g}\_{i\bar{\mathcal{g}}} \delta^{hm} R^{r}\_{rkm}\right) \\ &+ \frac{R}{(n-1)(n-2)} \left(\delta^{h}\_{k} \mathsf{g}\_{i\bar{j}} - \delta^{h}\_{j} \mathsf{g}\_{ik}\right),
\end{split}
$$

$$
\dot{\mathsf{C}}^{h}\_{ijk} = R^{h}\_{ijk} - \frac{R}{n(n-1)} \left(\delta^{h}\_{k} \mathsf{g}\_{i\bar{j}} - \delta^{h}\_{j} \mathsf{g}\_{ik}\right),
\tag{12}
$$

where *R<sup>h</sup> ijk*, *Rij* and *R* denote the curvature tensor, the Ricci tensor and the scalar curvature of Γ, respectively [6, 7].

In 1968, Yano and Sawaki defined and studied a new curvature tensor called quasi conformal curvature tensor on a Riemannian manifold [8]. Similarly, the notion of quasi conformal curvature tensor *W<sup>h</sup> ijk* of type 1, 3 ð Þ on a Weyl manifold of dimension *n* (*n*>3) is introduced by [9]

$$\mathcal{W}\_{ijk}^h = -(n-2)b\mathcal{C}\_{ijk}^h + [a + (n-2)b]\tilde{\mathcal{C}}\_{ijk}^h,\tag{13}$$

where *a, b* are arbitrary constants not simultaneously zero, *C<sup>h</sup> ijk* and *<sup>C</sup>*<sup>~</sup> *<sup>h</sup> ijk* are conformal curvature tensor and concircular curvature tensor of type 1, 3 ð Þ, respectively.

By substituting (11) and (12) in (13) the quasi conformal curvature tensor can be expressed by

$$\begin{split} \boldsymbol{W}\_{ijk}^{h} &= \boldsymbol{a}\boldsymbol{R}\_{ijk}^{h} + \boldsymbol{b} \left\{ \boldsymbol{\delta}\_{k}^{h}\boldsymbol{R}\_{ij} - \boldsymbol{\delta}\_{j}^{h}\boldsymbol{R}\_{ik} + \boldsymbol{g}\_{ij}\boldsymbol{g}^{hm}\boldsymbol{R}\_{mk} - \boldsymbol{g}\_{ik}\boldsymbol{g}^{hm}\boldsymbol{R}\_{mj} \right\} \\ &+ \frac{\boldsymbol{b}}{n} \left\{ (n-2)\boldsymbol{\delta}\_{i}^{h}\boldsymbol{R}\_{\gamma jk}^{r} + \boldsymbol{\delta}\_{j}^{h}\boldsymbol{R}\_{rki}^{r} - \boldsymbol{\delta}\_{k}^{h}\boldsymbol{R}\_{\gamma ji}^{r} + \boldsymbol{g}\_{ik}\boldsymbol{g}^{hm}\boldsymbol{R}\_{rjm}^{r} - \boldsymbol{g}\_{ij}\boldsymbol{g}^{hm}\boldsymbol{R}\_{rkm}^{r} \right\} \\ &- \frac{\boldsymbol{R}}{n} \left\{ \frac{\boldsymbol{a}}{n-1} + 2\boldsymbol{b} \right\} \left( \boldsymbol{\delta}\_{k}^{h}\boldsymbol{g}\_{ij} - \boldsymbol{\delta}\_{j}^{h}\boldsymbol{g}\_{ik} \right). \end{split}$$

#### **2. The concept of quasi conformally flatness on quasi Einstein-Weyl manifolds**

Quasi Einstein manifolds occupy a large place in the mathematical literature. For instance, research on quasi-Einstein manifolds helps us to understand the global character of topological spaces. Beside mathematics, studies on quasi-Einstein manifolds gain meaning with applications to general relativity.

The concept of quasi Einstein manifold was firstly introduced by M. C. Chaki and R. K. Maity as follows [10]:

A non-flat Riemannian manifold *Mn*, *gij* � � ð Þ *<sup>n</sup>*<sup>&</sup>gt; <sup>2</sup> is defined to be a quasi Einstein manifold if its Ricci tensor *Rij* of type 0, 2 ð Þ is not identically zero and satisfies the condition

$$R\_{i\bar{j}} = \alpha \mathbf{g}\_{i\bar{j}} + \beta A\_i A\_{\bar{j}},\tag{15}$$

where *α*, *β* are scalars of which *β* 6¼ 0 and *Ai* is a non-zero unit covariant vector field. In such an *n*-dimensional manifold which is denoted by ð Þ *QE <sup>n</sup>*; *α*, *β* are called associated scalars and *Ai* is called the generator of the manifold.

After Chaki and Maity, quasi Einstein manifolds are studied by many other authors. Moreover, in the articles [11–13] that inspired this study, conformal flatness and quasi conformal flatness were examined on quasi Einstein manifolds.

In this study, the concept of quasi conformal flatness on quasi Einstein manifolds were adapted to quasi Einstein-Weyl manifolds which was introduced by İ. Gül and E. Ö. Canfes as follows [14]:

**Definition 1.** *A non-flat Weyl manifold M*, *gij*, Φ*<sup>k</sup> of dimension n n*ð Þ <sup>&</sup>gt;<sup>2</sup> *is said to be a quasi Einstein-Weyl manifold if the symmetric part of its Ricci tensor Rij of type* (0,2) *is not identically zero and satisfies the condition*

$$R\_{(\vec{y})} = a \mathbf{g}\_{\vec{\imath}\vec{\jmath}} + \beta \mathbf{A}\_i \mathbf{A}\_{\vec{\jmath}} \tag{16}$$

*where α and β are scalars of weight* {�2} *with β* 6¼ 0*. The scalars α, β are called "associated scalars" and the unit covariant vector Ai of weight* {1} *is called "generator of the manifold". Such a manifold is denoted QEW* ð Þ*n.*

Therefore the aim of the present book chapter is to examine quasi conformally flat quasi Einstein-Weyl manifolds. It is organized as follows: In Section 1, the general information about Weyl manifolds are given. In Section 2, a theorem which shows the relationship between complementary vector field Φ*<sup>k</sup>* and generator *Ak* of quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* is proved and the expression of the curvature tensor of the quasi conformally flat quasi Einstein-Weyl manifold is obtained. In Section 3, three basic concepts are defined on quasi conformally flat quasi Einstein-Weyl manifolds and the necessary and sufficient conditions for these concepts are emphasized.

By means of (10) and (16), Ricci tensor *Rij* of ð Þ *QEW <sup>n</sup>* is expressed by

$$R\_{\vec{\eta}} = a \mathbf{g}\_{\vec{\eta}} + \beta A\_i A\_{\vec{\jmath}} + n \nabla\_i \Phi\_{\vec{\jmath}]} \tag{17}$$

which implies

$$R = am + \beta.\tag{18}$$

From (17), we have

$$\begin{split} R\_{\vec{\eta},l} - \mu\_l R\_{\vec{\eta}} &= \mathcal{g}\_{\vec{\eta}} \{ \alpha\_{\vec{\lambda}} - a(\mu\_l - 2\Phi\_l) \} + \left( \boldsymbol{\beta}\_{\vec{\lambda}} - \boldsymbol{\beta}\mu\_l \right) \mathbf{A}\_i \mathbf{A}\_j + \boldsymbol{\beta} \{ \mathbf{A}\_{i\vec{\lambda}} \mathbf{A}\_j + \mathbf{A}\_i \mathbf{A}\_{j,l} \} + \\ & \frac{\boldsymbol{\pi}}{2} \Big[ \left( \boldsymbol{\Phi}\_{\vec{\lambda}\vec{\mu}} - \boldsymbol{\Phi}\_{i\vec{\mu}} \right) - \mu\_l \left( \boldsymbol{\Phi}\_{\vec{\lambda}\vec{\mu}} - \boldsymbol{\Phi}\_{i\vec{\mu}} \right) \Big]. \end{split} \tag{19}$$

Since

$$\mathcal{R}\_{\downarrow} = \left(\mathbf{g}^{\vec{\eta}} \mathbf{R}\_{\vec{\eta}}\right)\_{\downarrow} = -\mathbf{2} \boldsymbol{\Phi}\_{\vec{\mathbf{k}}} \mathbf{g}^{\vec{\eta}} \mathbf{R}\_{\vec{\eta}} + \mathbf{g}^{\vec{\eta}} \mathbf{R}\_{\vec{\eta}\_{\downarrow}} \tag{20}$$

and *Ai* is normalized by the condition

$$\mathbf{g}^{\circ j} \mathbf{A}\_i \mathbf{A}\_j = \mathbf{1} \boldsymbol{\Leftrightarrow} \mathbf{A}^j \mathbf{A}\_j = \mathbf{1} \tag{21}$$

it is found that by multiplying (19) by *gij*

*Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds DOI: http://dx.doi.org/10.5772/intechopen.105683*

$$\mathcal{R}\_{\downarrow} - (\mu\_l - 2\Phi\_l)\mathcal{R} = n\{a\_{\downarrow} - a(\mu\_l - 2\Phi\_l)\} + (\beta\_{\downarrow} - \beta\mu\_l) + \beta(\mathcal{A}\_{i\downarrow}\mathcal{A}^i + \mathcal{A}^j\mathcal{A}\_{j\downarrow}).\tag{22}$$

By means of (18),

$$R\_{\downarrow} - \mu\_l R = n(a\_{\downarrow} - \mu\_l a) + (\beta\_{\downarrow} - \mu\_l \beta). \tag{23}$$

We obtain that

$$
\Phi\_l(\mathcal{R} - a\mathfrak{n}) = \beta \mathcal{A}\_{i,l} \mathcal{A}^i \tag{24}
$$

where *β* ¼ *R* � *αn*.

Hence we have the following:

**Theorem 1.** *The complementary vector field* Φ*<sup>i</sup> and the generator Ai of the quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup> are related by*

$$A\_{i,l}A^i = \Phi\_l. \tag{25}$$

Although the first part of the following example was given to prove the existence of the quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *n*>2 in [14], the verification of Theorem 1 is made by the author of the present book chapter in the second part of the example.

**Example 1**. *A three dimensional Weyl manifold M*<sup>3</sup> *is equipped with a metric gij by*

$$ds^2 = \mathbf{g}\_{\neq} d\mathbf{x}^i d\mathbf{x}^j = \mathbf{c}^{\mathbf{x}^1} \left[ \left(d\mathbf{x}^1\right)^2 + \left(d\mathbf{x}^2\right)^2 \right] + \left(d\mathbf{x}^3\right)^2$$

*and a 1-form* <sup>Φ</sup> *whose components* <sup>Φ</sup>*<sup>k</sup> given by* <sup>Φ</sup> <sup>¼</sup> *<sup>e</sup><sup>x</sup>*<sup>1</sup> *dx*<sup>2</sup> <sup>þ</sup> *dx*<sup>3</sup> *. The nonzero coefficients* Γ*<sup>i</sup> jk of a torsion-free connection* Γ *are [14]*

$$\begin{aligned} \Gamma^1\_{11} &= \frac{1}{2} \quad , \quad \Gamma^1\_{12} = \Gamma^1\_{21} = -e^{\mathbf{x}^1} \quad , \quad \Gamma^1\_{13} = \Gamma^1\_{31} = -\mathbf{1} \quad , \; \Gamma^1\_{22} = -\frac{1}{2} \; \; \vert \; \\ \Gamma^2\_{11} &= e^{\mathbf{x}^1} \quad , \; \Gamma^2\_{12} = \Gamma^2\_{21} = \frac{1}{2} \quad , \quad \Gamma^2\_{22} = -e^{\mathbf{x}^1} \quad , \; \Gamma^2\_{23} = \Gamma^2\_{32} = -\mathbf{1} \; \; \Gamma^2\_{33} = \mathbf{1} \quad , \; \\ \Gamma^3\_{11} &= \Gamma^3\_{22} = e^{\mathbf{x}^1} \quad , \; \Gamma^3\_{23} = \Gamma^3\_{32} = -e^{\mathbf{x}^1} \; \; \; \Gamma^3\_{33} = -\mathbf{1} \; \; \; . \end{aligned}$$

*It is clear that M*3, *gij*, Φ*<sup>k</sup>* � � *is a Weyl manifold with the connection* <sup>Γ</sup> *satisfying the condition (1). An elementary calculation gives the following nonzero components of the Ricci tensor [14]:*

$$\begin{aligned} R\_{11} &= \varepsilon^{\mathbf{x}^1} \left( \mathbf{1} + \varepsilon^{\mathbf{x}^1} \right) \ , \ R\_{12} = -R\_{21} = \frac{3}{2} \varepsilon^{\mathbf{x}^1} \ , \ R\_{22} = \varepsilon^{\mathbf{x}^1} \ , \ . \\\ R\_{23} &= R\_{32} = -\varepsilon^{\mathbf{x}^1} \ , \ . \ R\_{33} = \varepsilon^{\mathbf{x}^1} \ . \end{aligned}$$

*Moreover, the components of the symmetric parts of the Ricci tensor Rij and the scalar curvature R are [14]*

$$R\_{(11)} = e^{\mathbf{x}^1} \left( \mathbf{1} + e^{\mathbf{x}^1} \right) \,, \ R\_{(22)} = e^{\mathbf{x}^1} \,, \ R\_{(23)} = -e^{\mathbf{x}^1} \,, \ R\_{(33)} = e^{\mathbf{x}^1} \,, \ R = 2 \left( \mathbf{1} + e^{\mathbf{x}^1} \right) \,,$$

*Therefore, by considering (15), we find that [14]*

$$a = \mathbf{1} + e^{\mathbf{x}^1}, \quad \beta = -\left(\mathbf{1} + e^{\mathbf{x}^1}\right),$$

$$A\_1 = \mathbf{0} \quad A\_2 = \frac{e^{\mathbf{x}^1}}{\sqrt{\mathbf{1} + e^{\mathbf{x}^1}}} \quad , \ A\_3 = \frac{1}{\sqrt{\mathbf{1} + e^{\mathbf{x}^1}}}$$

*where gijAiAj* <sup>¼</sup> 1. *Thus, M*3, *gij*, <sup>Φ</sup>*<sup>k</sup>* � � *is a quasi-Einstein Weyl manifold. Now covariant derivatives of A*1, *<sup>A</sup>*<sup>2</sup> *and A*<sup>3</sup> *with respect to xk*ð Þ *<sup>k</sup>* <sup>¼</sup> 1, 2, 3 *are as follows:*

$$A\_{1,1} = 0 \quad , \ A\_{1,2} = 0 \quad , \ A\_{1,3} = 0;$$

$$A\_{2,1} = \frac{e^{\mathbf{x}^1}}{2(\mathbf{1} + e^{\mathbf{x}^1})^{\frac{3}{2}}}, \quad A\_{2,2} = \frac{e^{2\mathbf{x}^1} - e^{\mathbf{x}^1}}{\sqrt{\mathbf{1} + e^{\mathbf{x}^1}}}, \quad A\_{2,3} = \mathbf{0};\tag{26}$$

$$A\_{3,1} = \frac{-e^{\mathbf{x}^1}}{2(\mathbf{1} + e^{\mathbf{x}^1})^{\frac{3}{2}}}, \quad A\_{3,2} = \frac{2e^{\mathbf{x}^1}}{\sqrt{\mathbf{1} + e^{\mathbf{x}^1}}}, \quad A\_{3,3} = \frac{\mathbf{1} - e^{\mathbf{x}^1}}{\sqrt{\mathbf{1} + e^{\mathbf{x}^1}}}$$

*On the other hand, the reciprocals of Ai's are*

$$A^1 = 0 \quad A^2 = A^3 = \frac{1}{\sqrt{1 + \mathbf{e}^{x^1}}} \tag{27}$$

*By substituting (26) and (27) in (25),*

$$A\_{1,1}A^1 + A\_{2,1}A^2 + A\_{3,1}A^3 = \Phi\_1 \ , \ ,$$

$$A\_{1,2}A^1 + A\_{2,2}A^2 + A\_{3,2}A^3 = \Phi\_2 \ , \ ,$$

$$A\_{1,3}A^1 + A\_{2,3}A^2 + A\_{3,3}A^3 = \Phi\_3 \ .$$

*are obtained*.

A Weyl manifold *M*, *gij*, Φ*<sup>k</sup>* � � ð Þ *<sup>n</sup>*<sup>&</sup>gt; <sup>3</sup> is called quasi conformally flat, if the quasi conformal curvature tensor *W<sup>h</sup> ijk* satisfy the condition

$$\mathcal{W}^{h}\_{ijk} = \mathbf{0}.\tag{28}$$

Now, let us suppose that ð Þ *QEW <sup>n</sup>* ð Þ *n*> 3 is quasi conformally flat with *a* 6¼ 0 and *b* 6¼ 0. Then from (14),

$$R\_{ijk}^{b} = \frac{-b}{a} \left\{ \delta\_k^b R\_{ij} - \delta\_j^b R\_{ik} + \mathbf{g}\_{ij} \mathbf{g}^{hm} R\_{mk} - \mathbf{g}\_{ik} \mathbf{g}^{hm} R\_{mj} \right\} \tag{29}$$

$$+ \frac{-2b}{an} \left\{ (n-2)\delta\_i^b R\_{[bj]} + \delta\_j^b R\_{[ik]} - \delta\_k^b R\_{[j]} + \mathbf{g}\_{ik} \mathbf{g}^{hm} R\_{[mj]} - \mathbf{g}\_{ij} \mathbf{g}^{hm} R\_{[mk]} \right\}$$

$$+ \frac{R}{an} \left\{ \frac{a}{n-1} + 2b \right\} \left( \delta\_k^b \mathbf{g}\_{ij} - \delta\_j^b \mathbf{g}\_{ik} \right)$$

On the other hand, since it is assumed that the manifold is ð Þ *QEW <sup>n</sup>*, its Ricci tensor *Rij* can be written as (17) which satisfies (18).

Substituting (10), (17) and (18) in (29), the curvature tensor *R<sup>h</sup> ijk* is obtained as

$$R\_{ijk}^h = P\left(\delta\_{kb}^h \mathbf{g}\_{ij} - \delta\_{j}^h \mathbf{g}\_{ik}\right) + Q\left\{\delta\_k^h A\_i A\_j - \delta\_j^h A\_i A\_k + \mathbf{g}\_{ij} \mathbf{g}^{hm} A\_m A\_k - \mathbf{g}\_{ik} \mathbf{g}^{hm} A\_m A\_j\right\} \tag{30}$$

$$-\frac{(n-2)b}{a} \left\{2\delta\_i^h \nabla\_k \Phi\_{\hat{\boldsymbol{\beta}}} + \delta\_k^h \nabla\_i \Phi\_{\hat{\boldsymbol{\beta}}} - \delta\_{\hat{\boldsymbol{\beta}}}^h \nabla\_i \Phi\_{\hat{\boldsymbol{\beta}}} + \mathbf{g}\_{ij} \mathbf{g}^{hm} \nabla\_m \Phi\_{\hat{\boldsymbol{\beta}}} - \mathbf{g}\_{ik} \mathbf{g}^{hm} \nabla\_m \Phi\_{\hat{\boldsymbol{\beta}}}\right\}$$

where *<sup>P</sup>* <sup>¼</sup> *<sup>α</sup>n*þ*<sup>β</sup> n n*ð Þ �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*b<sup>β</sup> an* n o and *<sup>Q</sup>* <sup>¼</sup> �*<sup>b</sup> <sup>a</sup> β* are scalars. Ricci tensor *Rij* is obtained as

$$R\_{\vec{i}\vec{j}} = \{P(n-1) + Q\} \mathbf{g}\_{\vec{i}\vec{j}} + Q(n-2)A\_i A\_{\vec{j}} - \frac{n(n-2)b}{a} \nabla\_i \Phi\_{\vec{j}} \tag{31}$$

by contracting on the indices *h* and *k* in (30) and the scalar curvature is found in the form of

$$R = \{P(n-1) + Q\}n + Q(n-2) \tag{32}$$

by transvecting (31) by *gij:*

Using (10), (30), (31) and (32) in (11), it is obtained that

$$\mathbf{C}^{h}\_{ijk} = \mathbf{0} \tag{33}$$

leading us to following:

**Corollary 1.** *Quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *n* >3 *is conformally flat.*

#### **3. Some necessary and sufficient conditions on quasi conformally flat quasi Einstein-Weyl manifolds**

The concept of a space of quasi constant curvature was firstly introduced by Chen and Yano [15]. Similarly, we can define a Weyl manifold of quasi constant curvature as follows:

**Definition 2.** *A Weyl manifold M*, *gij*, Φ*<sup>k</sup>* � � ð Þ *<sup>n</sup>*><sup>3</sup> *is said to be of quasi constant curvature if it is conformally flat and its curvature tensor R<sup>h</sup> ijk of type* (1,3) *is in the form of*

$$R^h\_{\vec{\eta}k} = U\left(\delta^h\_{\mathsf{b}}\mathsf{g}\_{\vec{\eta}} - \delta^h\_j \mathsf{g}\_{ik}\right) + V\left\{\delta^h\_k A\_i A\_j - \delta^h\_j A\_i A\_k + \mathsf{g}\_{i\vec{\eta}} \mathsf{g}^{hm} A\_m A\_k - \mathsf{g}\_{i\vec{\eta}} \mathsf{g}^{hm} A\_m A\_j\right\}, \quad \text{(34)}$$

*where U and V are scalars with V* 6¼ 0 *and Ai is a covariant vector.*

On the other hand, Amur and Maralabhavi [16] proved that a quasi conformally flat Riemannian manifold is either conformally flat or Einstein. So, a quasi conformally flat quasi Einstein manifold, which is not Einstein, is conformally flat and its curvature tensor satisfies the condition in (32) with *a* 6¼ 0 and *b* 6¼ 0. Therefore, a quasi conformally flat quasi Einstein manifold with *a* 6¼ 0 and *b* 6¼ 0 is of quasi constant curvature.

However, the situation is more complicated for quasi conformally flat ð Þ *QEW <sup>n</sup>*. Because although quasi conformally flat ð Þ *QEW <sup>n</sup>* is conformally flat, it does not meet the requirement in (34) automatically. Therefore, a quasi conformally flat ð Þ *QEW <sup>n</sup>* will be of quasi constant curvature under special conditions.

Suppose that quasi conformally flat ð Þ *QEW <sup>n</sup>* ð Þ *n* >4 be of quasi constant curvature with the same definition in (2). Since *a* 6¼ 0 and *b* 6¼ 0, from (29),

$$2\delta\_i^h \nabla\_k \Phi\_{\hat{f}]} + \delta\_k^h \nabla\_i \Phi\_{\hat{f}]} - \delta\_{\hat{f}}^h \nabla\_i \Phi\_{\hat{k}]} + \mathcal{g}\_{i\hat{\mathcal{g}}} \mathbf{g}^{hm} \nabla\_m \Phi\_{\hat{k}]} - \mathcal{g}\_{i\hat{\mathcal{g}}} \mathbf{g}^{hm} \nabla\_m \Phi\_{\hat{f}]} = \mathbf{0} \tag{35}$$

is obtained. By transvecting (35) by *gij*,

$$
\nabla(n-4)\mathbf{g}^{hi}\nabla\_i\Phi\_{k\parallel} = \mathbf{0} \tag{36}
$$

and transvecting one more time by *ghj* with the assumption of *n*>4, it is found that

$$\nabla\_j \Phi\_{k\parallel} = \mathbf{0} \tag{37}$$

which means that the covariant derivative Φ*<sup>k</sup>*,*<sup>j</sup>* is symmetric.

Conversely, let the covariant derivative Φ*<sup>k</sup>*,*<sup>j</sup>* be symmetric in a quasi conformally flat ð Þ *QEW <sup>n</sup>* ð Þ *n* >4 *:* If (37) is substituted in (30), then (34) is obtained. Hence we get the following:

**Theorem 2.** *A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *n*>4 *to be of quasi constant curvature is that the covariant derivative* Φ*<sup>k</sup>*,*<sup>j</sup> is symmetric.*

Now, let us consider in which cases the covariant derivative Φ*<sup>k</sup>*,*<sup>j</sup>* is symmetric in a quasi conformally flat ð Þ *QEW <sup>n</sup>*, remembering that Φ*<sup>k</sup>* is different from zero or nongradient. So, let us give the definitions of some special vector fields in the Weyl manifold *M*, *gij*, Φ*<sup>k</sup>* :

**Definition 3.** *A vector field ξ in the Weyl manifold M*, *gij*, Φ*<sup>k</sup> is called torseforming if it satisfies the condition* ∇*Xξ* ¼ *ρX* þ *λ*ð Þ *X ξ, where ξ* ∈*χ*ð Þ *M , λ*ð Þ *X is a linear form and <sup>ρ</sup> is a function. In the local coordinates, it is expressed by* <sup>∇</sup>*iξ<sup>h</sup>* <sup>¼</sup> *ρδ<sup>h</sup> <sup>i</sup>* <sup>þ</sup> *<sup>ξ</sup><sup>h</sup>λi, where δ<sup>h</sup> <sup>i</sup> is the Kronecker symbol, ξ<sup>h</sup> and λ<sup>i</sup> are the components of ξ and λ. A torse-forming vector field <sup>ξ</sup> is called concircular if* <sup>∇</sup>*iξ<sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup>gij with <sup>ξ</sup><sup>j</sup>* <sup>¼</sup> *ghjξh.*

**Definition 4.** *A vector field ϕ in the Weyl manifold M*, *gij*, Φ*<sup>k</sup> is called <sup>ϕ</sup> (Ric) vector field if it satisfies* ∇*ϕ* ¼ *μRic, where μ is a constant and Ric is the Ricci tensor. In local coordinates, it is expressed by* ∇*iϕ<sup>j</sup>* ¼ *μRij, where ϕ<sup>i</sup> and Rij are the components of ϕ and Ric.*

**Definition 5.** *The components ϕ<sup>i</sup> of a vector field ϕ in the Weyl manifold M*, *gij*, Φ*<sup>k</sup> is defined as parallel if ϕ<sup>i</sup>* ,*<sup>j</sup>* <sup>¼</sup> <sup>0</sup> *and is defined concurrent if <sup>ϕ</sup><sup>i</sup>* ,*<sup>j</sup>* <sup>¼</sup> *<sup>c</sup>δ<sup>i</sup> j , where c is a constant.*

When we apply the above definitions to parallel, concurrent and concircular complementary vector field Φ*<sup>k</sup>* in a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>*, the covariant derivatives Φ*<sup>k</sup>*,*<sup>j</sup>* of these vector fields are

$$
\Phi\_{k\dot{\jmath}} = 2\Phi\_k \Phi\_{\dot{\jmath}}, \quad \Phi\_{k\dot{\jmath}} = 2\Phi\_k \Phi\_{\dot{\jmath}} + \mathcal{c} \mathbf{g}\_{k\dot{\jmath}}, \quad \Phi\_{k\dot{\jmath}} = \rho \mathbf{g}\_{k\dot{\jmath}}, \tag{38}
$$

*Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds DOI: http://dx.doi.org/10.5772/intechopen.105683*

respectively.

Now, let us consider Φ*<sup>k</sup>* as a *ϕ* (Ric) vector field. From Definition 3.4 and (10),

$$(\mathbf{1} - \mu \mathbf{n}) \nabla\_{\mathbf{j}} \Phi\_{\mathbf{k}]} = \mathbf{0}. \tag{39}$$

Finally, let us write the covariant derivative Φ*k*,*<sup>j</sup>* for a torse forming vector field Φ*<sup>k</sup>* defined by <sup>∇</sup>*i*Φ*<sup>h</sup>* <sup>¼</sup> *ρδ<sup>h</sup> <sup>i</sup>* <sup>þ</sup> <sup>Φ</sup>*hAi*, where *Ai* is the generator of ð Þ *QEW <sup>n</sup>*. By using Definition 3.3, we have

$$
\Phi\_{k\dot{j}} = 2\Phi\_{\dot{j}}\Phi\_k + \rho \mathbf{g}\_{k\dot{j}} + A\_{\dot{j}}\Phi\_k. \tag{40}
$$

By means of (38), (39) and (40), we can express the following:

**Corollary 2.** *A quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *n*> 4 *is of quasi constant curvature if the complementary vector field* Φ*<sup>k</sup> satisfies any one of the following:*

1.Φ*<sup>k</sup> is a parallel, concurrent or concircular vector field,*


Now, we seek a necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* to be recurrent. So, firstly, let us define the concept of recurrency in the quasi Einstein-Weyl manifolds by analogy to A.G. Walker's definition [17]:

**Definition 6.** *A non-flat Weyl manifold M*, *gij*, Φ*<sup>k</sup> is called recurrent if there exists a non-zero covariant vector μ<sup>l</sup> such that*

$$
\mathcal{R}^{h}\_{ijk,l} - \mu\_l \mathcal{R}^{h}\_{ijk} = \mathbf{0}.\tag{41}
$$

Suppose that quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *n*>3 , whose associated scalars *α* and *β* satisfy

$$\frac{a\_{\downarrow}}{a} = \frac{\beta\_{\downarrow}}{\beta} = \mu\_l - 2\Phi\_l,\tag{42}$$

is recurrent. From (39), it follows that

$$R\_{i\circ,l} - \mu\_l R\_{i\circ} = \mathbf{0} \tag{43}$$

by contracting on the indices *h* and *k* in (41) and transvecting (43) by *gij* gives us

$$R\_{,\ l} - (\mu\_l - 2\Phi\_l)R = \mathbf{0} \tag{44}$$

by means of (20).

By substituting (43) in (23), it is obtained that

$$n\left\{a\_{,l} - (\mu\_l - \mathfrak{A}\Phi\_l)a\right\} + \left\{\beta\_{,l} - (\mu\_l - \mathfrak{A}\Phi\_l)\beta\right\} = \mathbf{0} \tag{45}$$

which is satisfied by associated scalars in the above hypothesis. If (42) is substituted in (19) and transvecting by *A<sup>i</sup>*

$$\mathbf{0} = \beta \left( A\_{j,l} - \Phi\_l A\_j \right) + \frac{n}{2} \left[ \left( \Phi\_{j,il} - \Phi\_{i,jl} \right) - \mu\_l \left( \Phi\_{j,l} - \Phi\_{i,j} \right) \right] \mathbf{A}^i \tag{46}$$

is obtained. The conditions

$$A\_{j,l} - \Phi\_l A\_j = \mathbf{0} \Leftrightarrow A\_{j,l} = \Phi\_l A\_j \tag{47}$$

and

$$\left[\left(\Phi\_{j,il} - \Phi\_{i,jl}\right) - \mu\_l \left(\Phi\_{j,i} - \Phi\_{i,j}\right)\right] \mathbf{A}^i = \mathbf{0} \tag{48}$$

are satisfied in order to provide (46) since *β* 6¼ 0 and *n* >3 for a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>*.

If (47) is satisfied, then

$$\mathbf{A}^j\_{\;\!\!\!\!\!\!- \!\!\!- \!\!\!- \!\!\!\!- \!\!\!\!\!\!\/)} = -\Phi\_l \mathbf{A}^j,\tag{49}$$

where *Aj* <sup>¼</sup> *<sup>g</sup>jhAh*.

Let us first compute first and second covariant derivatives of the complementary vector Φ*<sup>j</sup>* defined by (25) by considering (21), (47) and (48):

$$
\Phi\_{j,i} = A^k A\_{k,i} - \Phi\_j \Phi\_i,\tag{50}
$$

$$\Phi\_{j,il} = 2\Phi\_j \Phi\_i \Phi\_l - A^k \left( A\_{k,il} \Phi\_j + A\_{k,jl} \Phi\_i + A\_{k,ji} \Phi\_l - A\_{k,jil} \right). \tag{51}$$

By using (50) and (51), the expressions <sup>Φ</sup>*<sup>j</sup>*,*il* � <sup>Φ</sup>*<sup>i</sup>*,*jl Ai* and *<sup>μ</sup><sup>l</sup>* <sup>Φ</sup>*<sup>j</sup>*,*<sup>i</sup>* � <sup>Φ</sup>*<sup>i</sup>*,*<sup>j</sup> Ai* in (48) can be written as

$$\left(\Phi\_{j,il} - \Phi\_{i,jl}\right)A^i = A^i A^k \left[\Phi\_l \left(A\_{k,ij} - A\_{k,ji}\right) + \left(A\_{k,jl} - A\_{k,jl}\right)\right],\tag{52}$$

$$
\mu\_l (\Phi\_{j,i} - \Phi\_{i,j}) A^i = A^i A^k \left[ \mu\_l (A\_{k, \circ i} - A\_{k, \circ j}) \right]. \tag{53}
$$

If (52) and (53) are substituted in (48), then

$$A^i A^k \left[ \left( A\_{k, \vec{\imath} \vec{l}} - A\_{k, \vec{\jmath} \vec{l}} \right) - \left( \mu\_l + \Phi\_l \right) \left( A\_{k, \vec{\imath} \vec{\imath}} - A\_{k, \vec{\imath} \vec{\jmath}} \right) \right] = \mathbf{0} \tag{54}$$

is obtained. Since *Ai* and *A<sup>k</sup>* 's are linearly independent,

$$A\_{k,jl} = (\mu\_l + \Phi\_l) A\_{k,jl}.\tag{55}$$

*Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds DOI: http://dx.doi.org/10.5772/intechopen.105683*

Conversely, let (47) and (55) be satisfied in a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* whose associated scalars *α* and *β* satisfying (42). From (30),

*Rh ijk*,*<sup>l</sup>* � *<sup>μ</sup>lR<sup>h</sup> ijk* <sup>¼</sup> f g *<sup>P</sup>*,*<sup>l</sup>* � ð Þ *<sup>μ</sup><sup>l</sup>* � <sup>2</sup>Φ*<sup>l</sup> <sup>P</sup> <sup>δ</sup><sup>h</sup> kgij* � *<sup>δ</sup><sup>h</sup> <sup>j</sup> gik* � � <sup>þ</sup> *<sup>Q</sup>*,*<sup>l</sup>* � *<sup>μ</sup>lQ* � � *<sup>δ</sup><sup>h</sup> kAiAj* � *<sup>δ</sup><sup>h</sup> <sup>j</sup> AiAk* <sup>þ</sup> *gijghmAmAk* � *gikghmAmAj* n o <sup>þ</sup> *<sup>Q</sup> <sup>δ</sup><sup>h</sup> <sup>k</sup> Ai*,*lAj* þ *AiAj*,*<sup>l</sup>* � � � *<sup>δ</sup><sup>h</sup> <sup>j</sup>* ð Þ *Ai*,*lAk* þ *AiAk*,*<sup>l</sup>* n <sup>þ</sup> *gijghm*ð*Am*,*lAk* <sup>þ</sup> *AmAk*,*l*Þ � *gikghm Am*,*lAj* <sup>þ</sup> *AmAj*,*<sup>l</sup>* � �o � ð Þ *<sup>n</sup>* � <sup>2</sup> *<sup>b</sup> a δh <sup>i</sup>* <sup>Φ</sup>*<sup>j</sup>*,*kl* � <sup>Φ</sup>*<sup>k</sup>*,*jl* � � � *<sup>μ</sup><sup>l</sup>* <sup>Φ</sup>*<sup>j</sup>*,*<sup>k</sup>* � <sup>Φ</sup>*<sup>k</sup>*,*<sup>j</sup>* � � � � ( þ 1 2 *δh <sup>k</sup>* <sup>Φ</sup>*<sup>j</sup>*,*il* � <sup>Φ</sup>*<sup>i</sup>*,*jl* � � � *<sup>μ</sup><sup>l</sup>* <sup>Φ</sup>*<sup>j</sup>*,*<sup>i</sup>* � <sup>Φ</sup>*<sup>i</sup>*,*<sup>j</sup>* � � � � � 1 2 *δh <sup>j</sup>* ½ � ð Þ� Φ*<sup>k</sup>*,*il* � Φ*<sup>i</sup>*,*kl μl*ð Þ Φ*<sup>k</sup>*,*<sup>i</sup>* � Φ*<sup>i</sup>*,*<sup>k</sup>* þ 1 2 *gijghm*½ � <sup>ð</sup>Φ*<sup>k</sup>*,*ml* � <sup>Φ</sup>*<sup>m</sup>*,*kl*Þ � *<sup>μ</sup>l*ð Þ <sup>Φ</sup>*<sup>k</sup>*,*<sup>m</sup>* � <sup>Φ</sup>*<sup>m</sup>*,*<sup>k</sup>* � 1 2 *gikghm* <sup>Φ</sup>*<sup>j</sup>*,*ml* � <sup>Φ</sup>*<sup>m</sup>*,*jl* � � � *<sup>μ</sup><sup>l</sup>* <sup>Φ</sup>*<sup>j</sup>*,*<sup>m</sup>* � <sup>Φ</sup>*<sup>m</sup>*,*<sup>j</sup>* � � � � ) (56)

If (42), (47) and (55) are written in (56), then (41) is obtained. Hence we can state the following:

**Theorem 3.** *A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *n*>3 *whose recurrent scalars α and β having the same recurrency vector μ<sup>l</sup>* � 2Φ*<sup>l</sup> to be recurrent is that the equations Ak*,*<sup>j</sup>* ¼ Φ*jAk and Ak*,*jil* ¼ ð Þ *μ<sup>l</sup>* þ Φ*<sup>l</sup> Ak*,*ji are satisfied.*

Let us dedicate the last part of this section to the concept of semi-symmetricness in a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>*. Firstly, let us define semi-symmetric ð Þ *QEW <sup>n</sup>* similar to the definition which is made by Szabo for Riemannian manifolds [18] as follows:

**Definition 7.** *A non-flat Weyl manifold M*, *gij*, Φ*<sup>k</sup>* � � *is called semi-symmetric if its curvature tensor R<sup>h</sup> ijk of type* (1,3) *satisfies the condition*

$$R\_{ijk,lm}^h - R\_{ijk,ml}^h = \mathbf{0}.\tag{57}$$

It follows that

$$R\_{\vec{ij},lm} - R\_{\vec{ij},ml} = \mathbf{0} \tag{58}$$

by contracting on the indices *h* and *k* in (57).

Let us suppose that a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *n*>3 with *aαn* 6¼ *bβ*ð Þ *n* � 2 is semi-symmetric. From (17) and (58),

$$\mathbf{0} = \beta \left[ (A\_{i,lm} - A\_{i,ml})A\_j + A\_i \left( A\_{j,lm} - A\_{j,ml} \right) \right] + 2\alpha \mathbf{g}\_{ij} (\Phi\_{l,m} - \Phi\_{m,l}) \tag{59}$$

$$+ \frac{n}{2} \left[ \left( \Phi\_{j,lm} - \Phi\_{i,jlm} \right) - \left( \Phi\_{j,ml} - \Phi\_{i,jml} \right) \right].$$

With the aid of the Ricci identity given as

$$
\upsilon\_{i,jk} - \upsilon\_{i,kj} = \upsilon\_h R^h\_{ijk},
\tag{60}
$$

where *vi*'s are the components of a covariant vector, it is obtained that

$$\begin{split} \mathbf{0} &= \quad \beta \Big[ \big( A\_h R\_{ilm}^h \big) A\_j + A\_i \big( A\_h R\_{jlm}^h \big) \Big] + 2 \alpha \mathbf{g}\_{ij} (\boldsymbol{\Phi}\_{l,m} - \boldsymbol{\Phi}\_{m,l}) \\ &+ \frac{n}{2} \big[ \big( \boldsymbol{\Phi}\_{j,lm} - \boldsymbol{\Phi}\_{i,jlm} \big) - \big( \boldsymbol{\Phi}\_{j,ml} - \boldsymbol{\Phi}\_{i,jml} \big) \Big]. \end{split} \tag{61}$$

If (61) is transvected by *gij*, it is found that

$$\mathbf{0} = \mathbf{2}\beta \Big(A\_h R\_{ilm}^h \big) A^i + \mathbf{2}\alpha n (\Phi\_{l,m} - \Phi\_{m,l}).\tag{62}$$

Substituting the following equation, resulted from (48),

$$(A\_h R\_{ilm}^h) A^i = -\frac{(n-2)b}{a} (\Phi\_{l,m} - \Phi\_{m,l}) \tag{63}$$

which is valid in a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* in (62) gives us

$$(\Phi\_{l,m} - \Phi\_{m,l})\left(an - \beta\frac{(n-2)b}{a}\right) = 0.\tag{64}$$

Because of the restriction on *α* and *β*,

$$
\Phi\_{l,m} - \Phi\_{m,l} = 0.\tag{65}
$$

If we form the difference Φ*<sup>l</sup>*,*<sup>m</sup>* � Φ*<sup>m</sup>*,*<sup>l</sup>* after taking covariant derivative of (25) with repect to *x<sup>m</sup>*, we have

$$
\Phi\_{l,m} - \Phi\_{m,l} = A^i (A\_{i,lm} - A\_{i,ml}) + A\_{i,l} A^i\_{,m} - A\_{i,m} A^i\_{,l}.\tag{66}
$$

If firstly rearranging the first term on the right hand side of the equation in (66) with the help of (60) and then using (63) and (64) in the resulting equation gives

$$A\_{i,l}A\_{,m}^i - A\_{i,m}A\_{,l}^i = \mathbf{0} \tag{67}$$

or equivalently

$$A\_{i,l}A\_{j,m} - A\_{j,l}A\_{i,m} = 0.\tag{68}$$

Conversely, let us assume that the generator *Ai* of a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *n* >3 satisfies the condition (67) or equivalently (68). If (67) is substituted in (66), then (65) is satisfied by means of (60). In this case,

$$A^i(A\_{i,lm} - A\_{i,ml}) = A^i(A\_h R^h\_{ilm}) = \mathbf{0} \tag{69}$$

*Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds DOI: http://dx.doi.org/10.5772/intechopen.105683*

which means that

$$A\_{i,lm} - A\_{i,ml} = 0\tag{70}$$

since *A<sup>i</sup>* 's are linearly independent. From (30),

*Rh ijk*,*ls* <sup>¼</sup> ð Þ *<sup>P</sup>*,*ls* <sup>þ</sup> <sup>2</sup>Φ*l*,*sP* <sup>þ</sup> <sup>2</sup>Φ*lP*,*<sup>s</sup>* <sup>þ</sup> <sup>2</sup>Φ*sP*,*<sup>l</sup>* <sup>þ</sup> <sup>4</sup>Φ*l*Φ*sP <sup>δ</sup><sup>h</sup> kgij* � *<sup>δ</sup><sup>h</sup> <sup>j</sup> gik* � � <sup>þ</sup>*Q*,*ls <sup>δ</sup><sup>h</sup> kAiAj* � *<sup>δ</sup><sup>h</sup> <sup>j</sup> AiAk* <sup>þ</sup> *gijghmAmAk* � *gikghmAmAj* n o <sup>þ</sup>*Q*,*<sup>l</sup> <sup>δ</sup><sup>h</sup> <sup>k</sup> Ai*,*sAj* þ *AiAj*,*<sup>s</sup>* � � � *<sup>δ</sup><sup>h</sup> <sup>j</sup>* <sup>ð</sup>*Ai*,*sAk* <sup>þ</sup> *AiAk*,*<sup>s</sup>*Þ þ *gijghm*ð Þ *Am*,*sAk* <sup>þ</sup> *AmAk*,*<sup>s</sup>* n o �*gikghm Am*,*sAj* <sup>þ</sup> *AmAj*,*<sup>s</sup>* � � <sup>þ</sup>*Q*,*<sup>s</sup> <sup>δ</sup><sup>h</sup> <sup>k</sup> Ai*,*lAj* þ *AiAj*,*<sup>l</sup>* � � � � *<sup>δ</sup><sup>h</sup> <sup>j</sup>* <sup>ð</sup>*Ai*,*lAk* <sup>þ</sup> *AiAk*,*<sup>l</sup>*Þ þ *gijghm*ð Þ *Am*,*lAk* <sup>þ</sup> *AmAk*,*<sup>l</sup>* �*gikghm Am*,*lAj* <sup>þ</sup> *AmAj*,*<sup>l</sup>* � �� <sup>þ</sup> *<sup>Q</sup> <sup>δ</sup><sup>h</sup> <sup>k</sup> Ai*,*lsAj* <sup>þ</sup> *Ai*,*lAj*,*<sup>s</sup>* <sup>þ</sup> *Ai*,*sAj*,*<sup>l</sup>* <sup>þ</sup> *AiAj*,*ls* � � � � �*δ<sup>h</sup> <sup>j</sup>* ð Þ *Ai*,*lsAk* þ *Ai*,*lAk*,*<sup>s</sup>* þ *Ai*,*sAk*,*<sup>l</sup>* þ *AiAk*,*ls* <sup>þ</sup>*gijghm*ð Þ *Am*,*lsAk* <sup>þ</sup> *Am*,*lAk*,*<sup>s</sup>* <sup>þ</sup> *Am*,*sAk*,*<sup>l</sup>* <sup>þ</sup> *AmAk*,*ls* �*gikghm Am*,*lsAj* <sup>þ</sup> *Am*,*lAj*,*<sup>s</sup>* <sup>þ</sup> *Am*,*sAj*,*<sup>l</sup>* <sup>þ</sup> *AmAj*,*ls* � �� (71)

is obtained. If necessary simplifications are made in the difference *R<sup>h</sup> ijk*,*ls* � *<sup>R</sup><sup>h</sup> ijk*,*sl* which is formed by means of (66), then it is found that

$$\begin{aligned} R\_{ijk,l}^h - R\_{ijk,sl}^h &= Q \left\{ \delta\_k^h \left[ (A\_{i,ls} - A\_{i,sl}) A\_j + A\_i (A\_{j,ls} - A\_{j,sl}) \right] \right. \\ &\quad - \delta\_j^h \left[ (A\_{i,ls} - A\_{i,sl}) A\_k + A\_i (A\_{k,ls} - A\_{k,sl}) \right] \\ &\quad + \mathbf{g}\_{ij} \mathbf{g}^{hm} \left[ (A\_{m,ls} - A\_{m,sl}) A\_k + A\_m (A\_{k,ls} - A\_{k,sl}) \right] \\ &\quad - \mathbf{g}\_{ik} \mathbf{g}^{hm} \left[ (A\_{m,ls} - A\_{m,sl}) A\_j + A\_m (A\_{j,ls} - A\_{j,sl}) \right] \right\} \end{aligned} \tag{72}$$

If (70) is written in (72), then we have

$$R\_{ijk,ls}^h - R\_{ijk,sl}^h = \mathbf{0} \tag{73}$$

which tells us that quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* is semi-symmetric. Therefore we can express the following:

**Theorem 4.** *A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *n*>3 *with αan* 6¼ *βb n*ð Þ � 2 *to be semi-symmetric is that the equation Ai*,*lAi* ,*<sup>m</sup>* � *Ai*,*mAi* ,*<sup>l</sup>* ¼ 0 *is satisfied.*

In the last part of this section, let us take a look at the relationships between to be of quasi constant curvature and to be semi-symmetric in a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>*.

If we combine Theorem 1 with Theorem 3 we get the following:

**Corollary 3.** *A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *n*>4 *with αan* 6¼ *βb n*ð Þ � 2 *to be semi-symmetric is that the manifold is of quasi constant curvature.*

Now, we will examine two special cases of the generator *Ai* of a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>*:

**I.case:** For a parallel generator *Ai* ; since

$$A\_{i,l} = 2A\_i \Phi\_l \text{ and} \\ A\_{i,lm} - A\_{i,ml} = 2A\_i(\Phi\_{l,m} - \Phi\_{m,l}) \tag{74}$$

from Definition 5, it is clear that a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *<sup>n</sup>*><sup>4</sup> , which the generator *Ai* is parallel, is automatically semisymmetric. If Definition 5 and the equations in (74) are used in (66), then (65) is obtained. This means that the manifold is of quasi constant curvature by means of Theorem 2.

Conversely, let us assume that quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *<sup>n</sup>* <sup>&</sup>gt;<sup>4</sup> , which the generator *Ai* is parallel, is of quasi constant curvature. In this case, if (70), which is implied by (64), is substituted in (72), then (73) is achieved which means that the manifold is semi-symmetric. Hence we can state the following:

**Theorem 5.** *A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *<sup>n</sup>*><sup>4</sup> *which the generator A<sup>i</sup> is parallel to be semisymmetric is that the manifold is of quasi constant curvature.*

**II.case:** Let us consider a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *<sup>n</sup>*><sup>4</sup> with 2*<sup>β</sup>* <sup>þ</sup> *<sup>α</sup><sup>n</sup>* 6¼ 0, which the generator *<sup>A</sup><sup>i</sup>* is concurrent, is semisymmetric. From Definition 5, it follows that

$$A\_{i,l} = 2A\_i \Phi\_l + c\mathbf{g}\_{il}, \qquad \qquad A\_{i,lm} - A\_{i,ml} = 2A\_i(\Phi\_{l,m} - \Phi\_{m,l}),$$

$$A\_{i,l}A\_{,m}^i - A\_{i,m}A\_{,l}^i = c(A\_{m,l} - A\_{l,m})\ . \tag{75}$$

If Definition 5 and the equations in (75) are used in (66), then

$$
\Phi\_{l,m} - \Phi\_{m,l} = c(A\_{l,m} - A\_{m,l}).\tag{76}
$$

Using (75) and (76) in (62) gives

$$c(4\beta + 2\alpha n)(A\_{l,m} - A\_{m,l}) = 0.\tag{77}$$

Because of the assumption on *α* and *β*, *Al*,*<sup>m</sup>* � *Am*,*<sup>l</sup>* ¼ 0 and therefore Φ*<sup>l</sup>*,*<sup>m</sup>* � Φ*<sup>m</sup>*,*<sup>l</sup>* ¼ 0 by (76) which tells us that the manifold is of quasi constant curvature.

Conversely, a quasi conformally flat quasi Einstein-Weyl manifold ð Þ *QEW <sup>n</sup>* ð Þ *<sup>n</sup>*><sup>4</sup> , which the generator *<sup>A</sup><sup>i</sup>* is concurrent, is of quasi constant curvature. Then, by Theorem 1, Φ*<sup>l</sup>*,*<sup>m</sup>* � Φ*<sup>m</sup>*,*<sup>l</sup>* ¼ 0 which is equivalent to *Al*,*<sup>m</sup>* � *Am*,*<sup>l</sup>* ¼ 0 by (76). If the last equation is substituted in (72), then (73) is obtained which means that the manifold is semi-symmetric. Hence we can state the following:

**Theorem 6.** *A necessary and sufficient condition for a quasi conformally flat quasi Einstein-Weyl manifold QEW* ð Þ*<sup>n</sup>* ð Þ *<sup>n</sup>*><sup>4</sup> *with* <sup>2</sup>*<sup>β</sup>* <sup>þ</sup> *<sup>α</sup><sup>n</sup>* 6¼ <sup>0</sup> *which the generator A<sup>i</sup> is concurrent to be semi-symmetric is that the manifold is of quasi constant curvature.*

#### **Acknowledgements**

The author is grateful to the referee for his/her valuable comments and suggestions for the improvement of the book chapter.

*Perspective Chapter: Quasi Conformally Flat Quasi Einstein-Weyl Manifolds DOI: http://dx.doi.org/10.5772/intechopen.105683*

### **Author details**

Fusun Nurcan Department of Mathematics, Marmara University, Istanbul, Turkey

\*Address all correspondence to: fnurcan622@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[6] Miron R. Mouvements conformes dans les espaces *Wn* and *Nn*. Tensor. 1968;**19**:33-41

[7] Ozdeger A, Senturk Z. Generalized Circles in Weyl Spaces and theirconformal mapping. Universitatis Debreceniensis. 2002;**60**:75-87

[8] Yano K, Sawaki S. Riemannian Manifolds admitting a conformal transformation group. Journal of Differential Geometry. 1968;**2**:161-184

[9] Nurcan F. Quasi conformally symmetric Weyl Manifolds. International Journal of Pure and Applied Mathematics. 2015;**3**:415-425

[10] Chaki MC, Maity RK. On quasi Einstein manifolds. Universitatis Debreceniensis. 2000;**57**:297-306. DOI: 10.1016/s0014-5793(01)03293-8

[11] De UC, Sengupta J, Saha D. Conformally flat quasi Einstein spaces. Kyungpook National University. 2006; **46**:417-423

[12] Shaikh AA, Patra A. On quasiconformally flat quasi-Einstein spaces. Differential Geometry-Dynamical Systems. 2010;**12**:201-212

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#### **Chapter 3**

## An Introduction to the Generalized Gauss-Bonnet-Chern Theorem

*Paul Bracken*

#### **Abstract**

This work studies the mathematical structures which are relevant to differentiable manifolds needed to prove the Gauss-Bonnet-Chern theorem. These structures include de Rham cohomology vector spaces of the manifold, characteristic classes such as the Euler class, pfaffians, and some fiber bundles with useful properties. The paper presents a unified approach that makes use of fiber bundles and leads to a noncomputational proof of the Gauss-Bonnet-Chern Theorem. It is indicated how it can be generalized to manifolds with boundary.

**Keywords:** manifold, Euler characteristic, bundle, fiber, projection, universal

#### **1. Introduction**

One of the great achievements of differential geometry is the Gauss-Bonnet theorem. In its original form, the theorem is a statement about surfaces which connect their geometry in the sense of curvature to the underlying topology of the space, in the sense of the Euler characteristic [1–5]. The most elementary case of the theorem states that the sum of the angles of a triangle in the plane is *π* radians. If the surface is deformed, the Euler characteristic does not vary as it is a topological invariant, while the curvature at certain points does change [6–8]. The theorem states that the total integral of the curvature remains the same, no matter how the deformation is performed. If there is a sphere with a ding, its total curvature is 4*π* since its Euler characteristic is two. This is the case no matter how big or deep the actual deformation is. A torus has Euler characteristic zero, so its total curvature must also be zero. If the torus carries the usual Riemannian metric from its embedding in ℝ3, then the inside has negative Gaussian curvature, and the outside has positive Gaussian curvature, so the total curvature is zero. It is not possible to specify a Riemannian metric on the torus which has everywhere positive or everywhere negative Gaussian curvature. Manifolds *M* have dimension *n* unless stated otherwise [9–11]. There are many applications of this theorem in both mathematics and mathematical physics such as in gravity [12–14], string theory [15] and even in the study of Ricci flow [16].

Although the curvature *K* is defined intrinsically in terms of the metric on the manifold *M*. It can also be defined for *n* ¼ 2 extrinsically when the metric on *M* is induced by an embedding *<sup>M</sup>* <sup>⊂</sup> <sup>ℝ</sup>3. In fact, it *<sup>ν</sup>* : *<sup>M</sup>* ! *<sup>S</sup>*<sup>2</sup> is the normal map and *da* is the volume element on *S*<sup>2</sup> , then *K d<sup>σ</sup>* <sup>¼</sup> *<sup>ν</sup>* <sup>∗</sup> *da* so that

$$\int\_M K d\sigma = \int\_M \nu^\* \, (da) = \deg(\nu) \cdot \int\_{S^2} da = 4\pi \cdot \deg(\nu). \tag{1}$$

Without bringing in differential geometric considerations, it is seen to be the case that degð Þ¼ *ν* ð Þ 1*=*2 *χ*ð Þ *M* , where *χ*ð Þ *M* is the Euler characteristic of *M*. Using this fact in (1), the Gauss-Bonnet theorem for a compact oriented surface *M*, the first version of the theorem is obtained for the case in which the metric on *M* arises by means of an embedding in ℝ<sup>3</sup>

$$\int\_{M} K d\sigma = 2\pi \chi(M). \tag{2}$$

It is the intention here to state and prove a general version of the theorem which applies to manifolds of even dimension, so a surface with *n* ¼ 2 is a special case. An intrinsic proof of the theorem was obtained by Chern 1944. The kind of argument outlined above was used by Hopf in developing the first generalization of the theorem. To outline the basic idea, consider a compact surface *M<sup>n</sup>* ⊂ ℝ*<sup>n</sup>*þ<sup>1</sup> when *n* is even. If *dμ<sup>g</sup>* is the volume form on the manifold and *dsn* denotes the volume element *S<sup>n</sup>*, then

$$\int\_{M^n} K\_n d\mu\_g = \int\_{M^n} \nu^\* \, ds\_n = \text{vol}(S^n) \cdot \text{deg}(\nu) = \frac{1}{2} \text{vol}(S^n) \cdot \chi(M^n). \tag{3}$$

This can be extended to any compact oriented Riemannian *<sup>n</sup>*-manifold *<sup>M</sup><sup>n</sup>* ð Þ , *<sup>g</sup>* which has even dimension, where *Kn* in a coordinate system is given by

$$K\_n = \frac{1}{\mathfrak{Z}^{n/2} n!} \sum\_{i\_1, \dots, i\_n} R\_{i\_1 i j\_1 j\_2} \cdots R\_{i\_{n-1} i\_n j\_{n-j\_n}} \frac{1}{\sqrt{\mathfrak{g}}} \,\epsilon^{j\_1 \dots j\_n} \cdot \frac{1}{\sqrt{\mathfrak{g}}} \,\epsilon^{j\_1 \dots j\_n} . \tag{4}$$

The ffiffi *g* p (4) is the square root of the determinant of the metric. With *Kn* given by (4), and *μ<sup>g</sup>* the volume form on the manifold, we are then led to conjecture that

$$\int\_{\mathcal{M}^n} K\_n \, d\mu\_{\mathcal{g}} = \frac{1}{2} \left[ \operatorname{vol}(\mathcal{S}^n) \cdot \chi(\mathcal{M}^n) \right],\tag{5}$$

where *M* is a compact, oriented Riemannian manifold with *n* even.

It is the objective to look at and study some of the ensuing developments which have led to a much deeper understanding of the foundations which underlie this theorem. It will be seen that this development leads to a completely noncomputational proof of this deep theorem.

#### **2. Characteristic classes**

When an oriented *n*-dimensional manifold *M*, *dμ<sup>g</sup>* � � is compact and closed, with *dμ<sup>g</sup>* is the volume form and *μ* the orientation of *M*, so every form has compact

support, Stokes theorem leads to the important theorem. Let *η* be any ð Þ *n* � 1 -form then

$$\int\_{\mathcal{M}} d\eta = \int\_{\partial \mathcal{M}} \eta. \tag{6}$$

Therefore an *n*-form *ω* on *M*, which is not exact, even though it must be closed as all *n*-forms on *M* are zero, can be found simply by locating an *ω* such that

$$\int\_{M} w \neq \mathbf{0}.\tag{7}$$

Such a form always exists, as it is known there is a form *ω* such that for *<sup>v</sup>*1, … , *vn* <sup>∈</sup> *Mp*, *<sup>ω</sup>*ð Þ *<sup>v</sup>*1, … , *vn* <sup>&</sup>gt; 0 if ½ �¼ *<sup>v</sup>*1, … , *vn <sup>μ</sup>p*. If *<sup>c</sup>* : ½ � 0, 1 *<sup>n</sup>* ! ð Þ *<sup>M</sup>*, *<sup>μ</sup>* preserves orientation, *<sup>c</sup>* <sup>∗</sup> *<sup>ω</sup>* on 0, 1 ½ �*<sup>n</sup>* is *g dx*<sup>1</sup> ∧⋯∧ *dx<sup>n</sup>* for some *<sup>g</sup>* <sup>&</sup>gt;0 on 0, 1 ½ �*<sup>n</sup>* , hence Ð *<sup>c</sup>ω* >0. This observation leads to this theorem. A smooth, oriented manifold is not smoothly contractible to a point. In fact, it is the shape of *M* not the size which determines whether or not every closed form on *M* is exact. More information about the shape of *M* can be obtained by analyzing more closely the extent to which closed forms are not exact. So how many non-exact *n*-forms are there on a compact oriented *n*-manifold If *ω* is not exact, the same holds for *ω* þ *dη* for *η* any ð Þ *n* � 1 -form *η*. Thus it is necessary to regard *ω* and *ω* þ *dη* as equivalent. This suggests an equivalence relation and directs one to think of this in terms of quotient spaces.

For each *k*, *Z<sup>k</sup>* ð Þ *M* denotes all closed *k*-forms on *M* and it is a vector space. The space *B<sup>k</sup>* ð Þ *<sup>M</sup>* of all exact *<sup>k</sup>*-forms is a subspace since *<sup>d</sup>*<sup>2</sup> <sup>¼</sup> 0. The quotient space is called the *k*-dimensional de Rham cohomology vector space of *M* and is defined to be

$$H^k(M) = Z^k(M) / \mathcal{B}^k(M). \tag{8}$$

The theorem of de Rham states that the vector space is isomorphic to a vector space defined just in terms of the topology of *M* called the *k*-dimensional cohomology group of *M* with real coefficients.

An element of *<sup>H</sup><sup>k</sup>*ð Þ *<sup>M</sup>* is an equivalence class ½ � *<sup>ω</sup>* of a closed form *<sup>ω</sup>* such that closed forms *ω*<sup>1</sup> and *ω*<sup>2</sup> are equivalent if and only if the difference is exact. In terms of these vector spaces, the Poincaré lemma gives *<sup>H</sup><sup>k</sup>* <sup>ℝ</sup>*<sup>n</sup>* ð Þ¼ 0, the vector space consisting of just the zero vector if *<sup>k</sup>*>0, or *<sup>H</sup><sup>k</sup>*ð Þ¼ *<sup>M</sup>* 0 if *<sup>M</sup>* is contractible and *<sup>k</sup>*>0. To compute *<sup>H</sup>*<sup>0</sup>ð Þ *<sup>M</sup>* note *<sup>B</sup>*<sup>0</sup>ð Þ¼ *<sup>M</sup>* 0 as there are no non-zero exact 0-forms as there are no nonzero minus one forms. Thus *<sup>H</sup>*<sup>0</sup>ð Þ *<sup>M</sup>* is the same as the vector space of all *<sup>C</sup>*<sup>∞</sup> functions *f* : *M* ! ℝ with *df* ¼ 0. If *M* is connected, this condition implies *f* is constant so *<sup>H</sup>*<sup>0</sup>ð Þ� *<sup>M</sup>* <sup>ℝ</sup> and its dimension is the number of components of *<sup>M</sup>*.

The de Rham cohomolgy vector spaces with compact support *H<sup>k</sup> <sup>c</sup>* ð Þ *M* are defined similarly to (8), that is, *H<sup>k</sup> <sup>c</sup>* ð Þ¼ *<sup>M</sup> <sup>Z</sup><sup>k</sup> <sup>c</sup>*ð Þ *<sup>M</sup> <sup>=</sup>Bk <sup>c</sup>*ð Þ *<sup>M</sup>* , where *<sup>Z</sup><sup>k</sup> <sup>c</sup>*ð Þ *M* is the vector space of all closed *k*-forms with compact support and *B<sup>k</sup> <sup>c</sup>*ð Þ *M* all *k*-forms *dη* where *η* is a *k*-form with compact support. If *M* is compact *H<sup>k</sup> <sup>c</sup>* ð Þ¼ *<sup>M</sup> <sup>H</sup><sup>k</sup>*ð Þ *<sup>M</sup>* .

Theorem 2.1. (The Poincaré-Duality Theorem) If *M* is a connected, oriented *n*manfold of finite type, then the map

$$\Pi: H^k(\mathcal{M}) \to H\_c^{n-k}(\mathcal{M}) \tag{9}$$

is an isomorphism for all *<sup>k</sup>*. □

This theorem eventually motivates the introduction of the Euler characteristic for any smooth connected oriented manifold *M*. Consider then a smooth *k*-dimensional vector bundle *ξ* ¼ *π* : *E* ! *M* over *M*. Orientations *μ* for *M*, and *ν* for *ξ* give an orientation *μ*⊕*ν* for the ð Þ *n* þ *k* -manifold *E*, since *E* is locally a product. Let f g *U*1, … , *Ur* be a cover of *M* by geodesically convex sets so small that each bundle *ξ* restricted to *Ui* is trivial. Then *<sup>π</sup>*�1ð Þ *<sup>U</sup>*<sup>1</sup> , … , *<sup>π</sup>*�1ð Þ *Ur* � � turns out to be a nice cover for *E*, so it is a manifold of finite type. For the section and projection maps *s*, *π*, *π*∘*s* ¼ *I* on *M* and *s*∘*π* is smoothly homotopic to the identity of *E*, so the map *π* <sup>∗</sup> : *H<sup>l</sup>* ð Þ! *<sup>M</sup> <sup>H</sup><sup>l</sup>* ð Þ *E* is an isomorphism for all *l*. The reason for mentioning (6) and Theorem 2.1 is that it shows there is a unique class *U* ∈ *H<sup>k</sup> <sup>c</sup>*ð Þ *E* such that

$$
\pi^\* \,:\, \mu \cup U = \mu \oplus \nu \subset H\_{\mathfrak{c}}^{n+k}(E). \tag{10}
$$

This class is called the Thom class of *ξ*.

A theorem states that if ð Þ *M*, *μ* is a compact oriented, connected manifold *ξ* ¼ *π* : *E* ! *M* an oriented *k*-plane bundle over *M* orientation *ν*, the Thom class *U* is the unique element of *H<sup>k</sup> <sup>c</sup>* ð Þ *E* such that for all *p* ∈ *M*, and *j <sup>p</sup>* : *Fp* ! *E* the inclusion map, we have *j* ∗ *<sup>p</sup> <sup>U</sup>* <sup>¼</sup> *<sup>ν</sup>p*. This condition has the implication that <sup>Ð</sup> ð Þ *Fp*,*ν<sup>p</sup> <sup>j</sup> <sup>p</sup>ω* ¼ 1, where *U* is the class of closed form *ω*.

The Thom class *U* of *ξ* ¼ *π* : *E* ! *M* can now be used to determine an element of *<sup>H</sup><sup>k</sup>*ð Þ *<sup>M</sup>* . Let *<sup>s</sup>* : *<sup>M</sup>* ! *<sup>E</sup>* be any section. There is always one, any two are clearly homotopic. Define the Euler class *<sup>χ</sup>*ð Þ *<sup>E</sup>* <sup>⊂</sup> *<sup>H</sup><sup>k</sup>* ð Þ *M* of *ξ* by

$$
\chi(\xi) = \mathfrak{s}^\* \ U. \tag{11}
$$

If *<sup>ξ</sup>* has a non-zero section *<sup>s</sup>* : *<sup>M</sup>* ! *<sup>E</sup>* and *<sup>ω</sup>* <sup>∈</sup>*C<sup>k</sup> <sup>c</sup>*ð Þ *M* represents *U*, a suitable multiple *c* � *s* of *s* takes *M* to the complement of support *ω*, so in this case, *χ ξ*ð Þ¼ ð Þ *<sup>c</sup>* � *<sup>s</sup>* <sup>∗</sup> *<sup>U</sup>* <sup>¼</sup> 0.

The term Euler class is connected with the special case of the bundle *TM* which has sections which are vector fields on *M*. If *X* is a vector field on *M* having an isolated zero at some point *p*, *X p*ð Þ¼ 0, but *X q*ð Þ 6¼ 0 for *q* 6¼ *p* in a neighborhood of *p*. An index of *X* at *p* can be defined. Suppose *X* is a vector field on an open set *U* ⊂ ℝ*<sup>n</sup>* with an isolated zero at 0 <sup>∈</sup> *<sup>U</sup>*. Define *<sup>f</sup> <sup>X</sup>* : *<sup>U</sup>* ! f g<sup>0</sup> ! *<sup>S</sup><sup>n</sup>*�<sup>1</sup> by *<sup>f</sup> <sup>X</sup>*ð Þ¼ *<sup>p</sup> X p*ð Þ*=*∣*X p*ð Þ∣. If *<sup>i</sup>* : *<sup>S</sup><sup>n</sup>*�<sup>1</sup> ! *<sup>U</sup>* is *i p*ð Þ¼ *<sup>ε</sup><sup>p</sup>* mapping *<sup>S</sup><sup>n</sup>*�<sup>1</sup> into *<sup>U</sup>*, then the map *<sup>f</sup> <sup>X</sup>*∘*<sup>i</sup>* : *<sup>S</sup><sup>n</sup>*�<sup>1</sup> ! *Sn*�<sup>1</sup> has a certain degree independent of *<sup>ε</sup>* for small *<sup>ε</sup>*, since maps *<sup>i</sup>*1, *<sup>i</sup>*<sup>2</sup> : *<sup>S</sup><sup>n</sup>*�<sup>1</sup> ! *<sup>U</sup>* correspinding to *ε*1, *ε*<sup>2</sup> will be smoothly homotopic. This degree is called the index of *X* at 0. Consider a diffeomorphism *<sup>h</sup>* : *<sup>U</sup>* ! *<sup>V</sup>* <sup>⊂</sup> <sup>ℝ</sup>*<sup>n</sup>* with *<sup>h</sup>*ð Þ¼ <sup>0</sup> 0, so *<sup>h</sup>*<sup>∗</sup> *<sup>X</sup>* is the vector field on *<sup>V</sup>* such that ð Þ *h*<sup>∗</sup> *X* ð Þ¼ *y h*<sup>∗</sup> *Xh*�<sup>1</sup> ð Þ*y* � �. So 0 is an isolated zero of *<sup>h</sup>*∗*X*. It can be shown, if *<sup>h</sup>* : *<sup>U</sup>* ! *<sup>V</sup>* <sup>⊂</sup> <sup>ℝ</sup>*<sup>n</sup>* is a diffeomorphism with *<sup>h</sup>*ð Þ¼ <sup>0</sup> 0 and *<sup>X</sup>* has an isolated zero at 0, the index of *h*∗*X* at 0 equals the index of *X* at 0.

As a consequence of this, an index of a vector field on a mainifold can be defined. If *X* is a vector field on *M*, with isolated zero at *p* ∈ *M*, choose a coordinate system ð Þ *x*, *U* such that *x p*ð Þ¼ *p* and define the index of *X* at *p* to be the index of *x*∗*X* at 0.

Theorem 2.2. Let *M* be a compact, connected manifold with orientation *μ*, also an orientation for the tangent bundle *ξ* ¼ *π* : *TM* ! *M*. Let *X* : *M* ! *TM* be a vector field with only a finite number of zeros and let *σ* be the sum of indices of *X* at these zeros. Then

*An Introduction to the Generalized Gauss-Bonnet-Chern Theorem DOI: http://dx.doi.org/10.5772/intechopen.105716*

$$\chi(\xi) = \sigma \cdot \mu \in H^0(\mathbb{M}). \tag{12}$$

□

#### **3. Pfaffians**

An intrinisic expression along with one in a coordinate system for the function *Kn* on a compact, oriented Riemannian manifold of even dimension has been given already. Another more important way of expressing *Kn* involves the curvature form Ω*<sup>i</sup> j* for a positively oriented orthonormal moving frame *X*1, … , *Xn* on *M*. In terms of these forms, the *n*-form *Kn dμg*, the one to be integrated, can be written down. A sum over permutations such as

$$\sum\_{\pi \in \mathcal{S}\_n} B(X\_{\pi(1)}, \dots, X\_{\pi(n)}),\tag{13}$$

can be written just as well as

$$\sum\_{j\_1,\ldots,j\_n} \varepsilon^{j\_1,\ldots,j\_n} B(\mathbf{X}\_{j\_1},\ldots,\mathbf{X}\_{j\_n}).\tag{14}$$

Suppose this is the *n*-fold wedge product

$$
\Omega\_{i\_2}^{i\_1} \wedge \dots \wedge \Omega\_{i\_n}^{i\_{n-1}}.\tag{15}
$$

Since the Ω*<sup>i</sup> <sup>j</sup>* are 2-forms, using the definition of wedge product,

$$\Omega\_{i\_2}^{j\_1} \wedge \dots \wedge \Omega\_{i\_n}^{j\_{n-1}}(X\_1, \dots, X\_n) = \frac{(2 + \cdots + 2)!}{2! \cdots 2!} \cdot \frac{1}{n!} \sum\_{j\_1, \dots, j\_n} e^{j\_1 \cdots i\_p} \, \Omega\_{i\_2}^{j\_1}(X\_{j\_1}, X\_{j\_2}) \cdots \Delta\_{i\_n}^{j\_{n-1}}(X\_{j\_{n-1}}, X\_{j\_n})$$

$$= \frac{1}{2^{n/2}} \sum\_{j\_1, \dots, j\_n} e^{j\_1 \cdots \dots j\_n} \langle \mathcal{R}(\mathbf{X}\_{j\_1}, \mathbf{X}\_{j\_2}) \mathbf{X}\_{j\_1}, \mathbf{X}\_{j\_1} \rangle \cdots \langle \mathcal{R}\left(\mathbf{X}\_{j\_{n-1}}, \mathbf{X}\_{j\_n}\right) \mathbf{X}\_{j\_n}, \mathbf{X}\_{j\_{n-1}} \rangle$$

$$= \frac{1}{2^{n/2}} \sum\_{j\_1, \dots, j\_n} R\_{i\_1 i\_2 j\_1} \cdots R\_{i\_{n-1} i\_j j\_{n-j} j\_n}. \tag{16}$$

Comparing this to (4), it may be concluded that

$$K\_n = \frac{1}{2^{n/2}n!} \sum\_{i\_1,\cdots,i\_n} \varepsilon^{i\_1,\cdots,i\_n} 2^{n/2} \mathfrak{Q}\_{i\_2}^{i\_1} \wedge \cdots \wedge \mathfrak{Q}\_{i\_n}^{i\_{n-1}} (X\_1,\cdots,X\_n). \tag{17}$$

When (17) is multiplied by the volume form *dμg*, it becomes

$$K\_n d\mu\_g = \frac{1}{n!} \sum\_{i\_1, \dots, i\_n} \Omega\_{i\_1}^{i\_1} \wedge \dots \wedge \Omega\_{i\_n}^{i\_{n-1}}.\tag{18}$$

By (18) the form on the right does depend on the choice of the positively oriented orthonormal frame, *X*1, … , *Xn*. There is a direct way to get this algebraically.

Suppose *<sup>A</sup>* is an *<sup>n</sup>* � *<sup>n</sup>* matrix *<sup>A</sup>* <sup>¼</sup> *aij* � � with *<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* even. Define the Pfaffian, Pfð Þ *A* of *A* to be

$$\text{Pf}(A) = \frac{1}{2^m m!} \sum\_{i\_1, \dots, i\_n} a\_{i\_1 i\_2} \cdots a\_{i\_{n-1} i\_n} \,. \tag{19}$$

Note that *<sup>ε</sup>i*1⋯*in* does not change when any permutation of the pairs *<sup>i</sup>*2*l*�<sup>1</sup> ð Þ , *<sup>i</sup>*2*<sup>l</sup>* . For any set *S* ¼ f g ð Þ *h*1, *k*<sup>1</sup> , ⋯,ð Þ *hm*, *km* of pairs of integers between 1 and *n*, let us define

$$
\varepsilon(\mathbf{S}) = \varepsilon^{h\_1 k\_1 \cdots h\_m k\_m}.\tag{20}
$$

It is not necessary to specify any ordering of pairs *hi* ð Þ , *ki* in *S*. Also a permutation of the pairs *i*2*l*�<sup>1</sup> ð Þ , *i*2*<sup>l</sup>* does not change the factor *ai*1*i*2⋯*ain*�<sup>1</sup>*in* . So for each *P* above define *aS* ¼ *ah*1*k*1⋯*ahmkm* . If *P* is the collection of all such *S*, we clearly have

$$\text{Pf}(A) = \frac{1}{\mathfrak{Z}^m} \sum\_{\mathcal{S} \in P} \varepsilon(\mathcal{S}) \mathfrak{a}\_{\mathcal{S}}.\tag{21}$$

Theorem 3.1. Let *n* ¼ 2*m* then for all *n* � *n* matrices *A* and *B*,

$$\text{Pf}\,(\mathcal{B}^{\ell}AB) = (\det \mathcal{B}) \cdot \text{Pf}\,(A). \tag{22}$$

and *Bt* denotes the transpose. If *<sup>B</sup>*<sup>∈</sup> *SO n*ð Þ then

$$\text{Pf}\left(B^{-1}AB\right) = \text{Pf}(A). \tag{23}$$

*Proof*:

$$\begin{split} \mathfrak{L}^{m} \cdot m! \, \mathrm{Pf}(B^{t}AB) &= \sum\_{i\_{1} \cdots i\_{n}} e^{i\_{1} \cdots i\_{n}} \sum\_{j\_{1} \cdots j\_{n}} \langle b\_{j\_{1} i\_{1}} a\_{j\_{1} j\_{2}} b\_{j\_{2} i\_{1}} \rangle \cdots \langle b\_{j\_{n-1} i\_{n-1}} a\_{j\_{n} \cdots j\_{n}} b\_{j\_{n} i\_{n}} \rangle \\ &= \sum\_{j\_{1} \cdots j\_{n}} \left[ \sum\_{j\_{1} \cdots i\_{n}} e^{i\_{1} \cdots i\_{n}} b\_{j\_{1} i\_{1}} \cdots b\_{j\_{n} i\_{n}} \right] a\_{j\_{1} j\_{2}} \cdots a\_{j\_{n} \cdots j\_{n}} = \sum\_{j\_{1}, \cdots, j\_{n}} e^{i\_{1} \cdots \cdots j\_{n}} \det(B) a\_{j\_{1} j\_{2}} \cdots a\_{j\_{n} \cdots j\_{n}} \\ &= \mathfrak{L}^{m} m! (\det B) \, \mathrm{Pf}(A) . \end{split} \tag{24}$$

□

This theorem was stated for matrices of real numbers, but Pfð Þ *A* can be defined provided the entries of *A* are in some commutative algebra over ℝ.

Consider again a positively oriented orthogonal moving frame *X*1, … ,*Xn* on *M*, with curvature forms Ω*<sup>i</sup> j* . For each *p*∈ *M*, the direct sum *A* ¼ ℝ⊕Ω<sup>2</sup> *Mp* � �⊕Ω<sup>2</sup> *Mp* � �⊕⋯ is a commutative algebra over ℝ under the operation ∧ . So one can consider Pfð Þ Ωð Þ *p* , where Ωð Þ *p* is an *n* � *n* matrix of connection 2-forms at *p*

$$\text{Pf}\left(\Omega\_p\right) = \frac{1}{2^m m!} \sum\_{i\_1,\dots,i\_n} \epsilon^{i\_1,\dots,i\_n} \Omega\_{i\_1}^{i\_1} \wedge \dots \wedge \Omega\_{i\_n}^{i\_{n-1}}(p). \tag{25}$$

If *X*<sup>0</sup> ¼ *X* � *a* is another positively oriented orthonormal moving frame then *a p*ð Þ<sup>∈</sup> *O n*ð Þ and the corresponding curvature forms satisfy <sup>Ω</sup><sup>0</sup> <sup>¼</sup> *<sup>a</sup>*�<sup>1</sup>Ω*a*. Then Theorem 3.1 implies that

*An Introduction to the Generalized Gauss-Bonnet-Chern Theorem DOI: http://dx.doi.org/10.5772/intechopen.105716*

$$\text{Pf}(\Omega'(p)) = \text{Pf}\left(a^{-1}(p)\Omega a(p)\right) = \text{Pf}(\Omega(p))\tag{26}$$

so the form P *<sup>i</sup>*1,⋯,*in <sup>ε</sup>i*1,⋯,*in* <sup>Ω</sup>*i*<sup>1</sup> *<sup>i</sup>*<sup>2</sup> ∧⋯∧ <sup>Ω</sup>*in*�<sup>1</sup> *in* is well defined.

#### **4. Bundles of paticular importance**

Projective *<sup>n</sup>*-space <sup>ℙ</sup>*<sup>n</sup>* can be defined as the set of all pairs ð Þ �*p*, *<sup>p</sup>* for *<sup>p</sup>* <sup>∈</sup>*Sn* <sup>⊂</sup> <sup>ℝ</sup>*n*þ<sup>1</sup> or the set of line through 0 in ℝ*n*þ<sup>1</sup> , since each lines intersects *S<sup>n</sup>* through two antipodal points. A Grassmannian manifold *Gn* <sup>ℝ</sup>*<sup>n</sup>* ð Þ is the set of all *<sup>n</sup>*-dimensional subspaces of ℝ*<sup>N</sup>* with *N* > 0. Over the Grassmannian manifold *Gn* ℝ*<sup>N</sup>* � �, there is a natural *<sup>n</sup>*-dimensional bundle *<sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>n</sup>* ð Þ constructed as follows. The total space of the bundle *<sup>E</sup> <sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � � � is the subset of *Gn* <sup>ℝ</sup>*<sup>N</sup>* � � � <sup>ℝ</sup>*<sup>N</sup>* consisting of all pairs

$$(\mathcal{W}, w) \in \mathcal{G}\_n \left(\mathbb{R}^N\right) \times \mathbb{R}^M, \qquad w \in \mathcal{W}. \tag{27}$$

The projection map which takes *<sup>E</sup> <sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � � � ! *Gn* <sup>ℝ</sup>*<sup>N</sup>* � � is *<sup>π</sup>*ð Þ¼ ð Þ *<sup>W</sup>*, *<sup>w</sup> <sup>W</sup>*. The fibre *<sup>π</sup>*�<sup>1</sup>ð Þ *<sup>W</sup>* over *<sup>W</sup>* of *Gn* <sup>ℝ</sup>*<sup>N</sup>* � � will be *<sup>W</sup>* itself or, more explicitly, f g ð Þ *<sup>W</sup>*, *<sup>w</sup>* : *<sup>w</sup>* <sup>∈</sup>*<sup>W</sup>* . A vector space structure is defined on *<sup>π</sup>*�<sup>1</sup>ð Þ *<sup>W</sup>* by using the vector space structure on *<sup>W</sup>* <sup>⊂</sup> <sup>ℝ</sup>*<sup>N</sup>*; if *<sup>a</sup>* is a scalar, then ð Þþ *<sup>W</sup>*, *<sup>w</sup>*<sup>1</sup> ð Þ¼ *<sup>W</sup>*, *<sup>w</sup>*<sup>2</sup> ð Þ *<sup>W</sup>*, *<sup>w</sup>*<sup>1</sup> <sup>þ</sup> *<sup>w</sup>*<sup>2</sup> and *a W*ð Þ¼ , *<sup>w</sup>* ð Þ *<sup>W</sup>*, *aw* . Also *<sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � satisfies the local triviality condition.

For *<sup>M</sup>* <sup>&</sup>gt; *<sup>N</sup>* there is a natural map *<sup>α</sup>* : *Gn* <sup>ℝ</sup>*<sup>N</sup>* � � ! *Gn* <sup>ℝ</sup>*<sup>M</sup>* � �, as an *<sup>n</sup>*-dimensional subspace of ℝ*<sup>N</sup>* can be considered an *n*-dimensional subspace of ℝ*<sup>M</sup>*. There is clearly a map *<sup>α</sup>* : *<sup>E</sup> <sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � � � ! *<sup>E</sup> <sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>M</sup>* � � � � such that ð Þ *<sup>α</sup>*, *<sup>α</sup>* is a bundle map from *<sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � to *ζ<sup>n</sup>* ℝ*<sup>M</sup>* � � and thus *ζ<sup>n</sup>* ℝ*<sup>N</sup>* � �≂*α*<sup>∗</sup> *ζ<sup>n</sup>* ℝ*<sup>M</sup>* � � � � .

In algebraic topology, one often considers the union *<sup>G</sup>*<sup>0</sup> <sup>ℝ</sup><sup>∞</sup> ð Þ of the increasing sequence *Gn* <sup>ℝ</sup>*<sup>n</sup>*þ<sup>1</sup> � �<sup>⊂</sup> *Gn* <sup>ℝ</sup>*<sup>n</sup>*þ<sup>1</sup> � � ⊂ ⋯ with weak topology; that is, a set *<sup>U</sup>* <sup>∈</sup> *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ¼ ∪*<sup>l</sup> Gn* ℝ*<sup>n</sup>*þ*<sup>l</sup>* � � is open if and only if *U*∩*Gn* ℝ*<sup>n</sup>*þ*<sup>l</sup>* � � is open in *Gn* ℝ*<sup>n</sup>*þ*<sup>l</sup>* � � for all *l*. There is a natural *<sup>n</sup>*-dimensional bundle *<sup>ζ</sup><sup>n</sup>* over *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ defined in a way similar to *<sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � such that the following properties are maintained: ð Þ*i* for every bundle *ξ* over a paracompact space *<sup>X</sup>*, there is a map *<sup>f</sup>* : *<sup>X</sup>* ! *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ such that *<sup>ξ</sup>*<sup>≃</sup> *<sup>f</sup>* <sup>∗</sup> *<sup>ζ</sup><sup>n</sup>* ð Þ. ð Þ *ii* if *<sup>f</sup>* <sup>0</sup>, *<sup>f</sup>* <sup>1</sup> : *<sup>X</sup>* ! *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ are maps of a paracompact space *<sup>X</sup>* into *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ with *<sup>f</sup>* <sup>∗</sup> 0 : *ζ<sup>n</sup>* ≃*f* <sup>∗</sup> <sup>1</sup> *ζ<sup>n</sup>* then *f* <sup>1</sup> ≃*f* <sup>0</sup>.

For this reason *<sup>ζ</sup><sup>n</sup>* is called the *universal n-dimensional bundle* and *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ, is called the classifying space for *n*-dimensional bundles since equivalence classes of *n*-dimensional bundles over *<sup>X</sup>* are classified by homotopy classes of maps of *<sup>X</sup>* into *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ. Now *Gn* <sup>ℝ</sup><sup>∞</sup> ð Þ is not a manifold so we continue to use the bundles *<sup>ζ</sup><sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � �, which are usually called universal bundles.

An orientation for a vector space *V* is an equivalence class of ordered bases for *V* where ð Þ� *<sup>v</sup>*1, … , *vn* ð Þ *<sup>w</sup>*1, … , *wn* if and only if *aij* � � defined by *wi* <sup>¼</sup> <sup>P</sup> *<sup>j</sup> aji vj* has det *aij* � �>0. There are only two such equivalence classes *<sup>η</sup>* and �*η*. An oriented vector space is a pair ð Þ *V*, *η* , where *η* is an orientation for *V*.

An orientation for a bundle *ξ* ¼ *π* : *E* ! *X* is a collection *η* ¼ f g *η<sup>x</sup>* of orientations for the fibres *<sup>π</sup>*�<sup>1</sup>ð Þ *<sup>x</sup>* which satisfy an obvious compatibility requirement, while an oriented bundle is a pair ð Þ *ξ*, *η* , where *η* is an orientation for *ξ*. Orientation *η* of *ξ* gives another �*η* ¼ �f g*η* if *X* is connected. This is the only other one for *ξ*. Define *ξ*1⊕*ξ*2, *μ*1⊕*μ*<sup>2</sup> ð Þ to be the sum *ξ*1⊕*ξ*<sup>2</sup> with the indicated orientation.

Suppose *ξ* ¼ *π* : *E* ! *M* is a smooth oriented *n*-dimensional vector bundle over a smooth manifold *<sup>M</sup>* of any dimension. The Euler class *χ ξ*ð Þ<sup>∈</sup> *<sup>H</sup>n*ð Þ *<sup>M</sup>* was defined by first defining the Thom class *<sup>U</sup>*ð Þ*<sup>ξ</sup>* <sup>∈</sup> *<sup>H</sup><sup>n</sup> <sup>c</sup>* ð Þ *E* . It can be proved *U*ð Þ*ξ* is the unique class whose restriction to each *<sup>π</sup>*�1ð Þ *<sup>p</sup>* is the generator *<sup>ν</sup><sup>p</sup>* <sup>¼</sup> *<sup>H</sup><sup>n</sup> <sup>c</sup> <sup>π</sup>*�<sup>1</sup> ð Þ ð Þ *<sup>p</sup>* determined by the orientation. This result leads directly into the next theorem.

*Theorem 4.1*. Let *ξ* ¼ *π* : *E* ! *M* be a smooth manifold where *M*<sup>0</sup> is also a compact manifold. If *<sup>E</sup>* is the total space of *<sup>f</sup>* <sup>∗</sup> *<sup>ξ</sup>* and <sup>~</sup>*<sup>f</sup>* : *<sup>E</sup>*<sup>0</sup> ! *<sup>E</sup>* is a bundle map covering *<sup>f</sup>*,

$$
\tilde{f}^\*(U(\xi)) = U(f^\*\xi) \in H\_c^n(E'). \tag{28}
$$

**Proof**: Note ~*f* has the property inverse of a compact set is compact, so ~*f* ∗ takes *Hn <sup>c</sup>* ð Þ *<sup>E</sup>* to *<sup>H</sup><sup>n</sup> <sup>c</sup> <sup>E</sup>*<sup>0</sup> ð Þ. Let *<sup>f</sup>* <sup>∗</sup> *<sup>ξ</sup>* be *<sup>π</sup>*<sup>0</sup> : *<sup>E</sup>*<sup>0</sup> ! *<sup>M</sup>*<sup>0</sup> . If *<sup>p</sup>*<sup>∈</sup> *<sup>M</sup>*<sup>0</sup> is any point, and *ip*<sup>0</sup> : *<sup>π</sup>*0�<sup>1</sup>ð Þ! *<sup>p</sup> <sup>E</sup>*<sup>0</sup> is the inclusion map, then

$$i\_p^\* \tilde{f} \, U(\xi) = \left(\tilde{f} \circ i\_{p'}\right)^\* U(\xi). \tag{29}$$

Recall how *f* <sup>∗</sup> *ξ* is defined then ~*f*∘*ip*<sup>0</sup> <sup>∗</sup> *<sup>U</sup>*ð Þ*<sup>ξ</sup>* must be the generator of *<sup>H</sup><sup>n</sup> <sup>c</sup> <sup>π</sup>*0�<sup>1</sup> *<sup>p</sup>*<sup>0</sup> ð Þ ð Þ , since *i* <sup>∗</sup> *f p*<sup>0</sup> ð Þ*U*ð Þ*<sup>ξ</sup>* is the generator of *<sup>H</sup><sup>n</sup> <sup>c</sup> <sup>π</sup>*�<sup>1</sup> *f p*<sup>0</sup> <sup>ð</sup> ð Þ ð Þ . This shows <sup>~</sup>*<sup>f</sup>* ∗ *<sup>U</sup>*ð Þ*<sup>ξ</sup>* must be *U f* <sup>∗</sup> ð Þ*<sup>ξ</sup>* . □

The Euler class *χ ξ*ð Þ was defined as *<sup>s</sup>* <sup>∗</sup> *<sup>U</sup>*ð Þ*<sup>ξ</sup>* for any section *<sup>s</sup>* of *<sup>ξ</sup>*. Suppose *<sup>s</sup>* <sup>¼</sup> 0 is the zero section, which is chosen. It can be shown that

$$f^\*\chi(\xi) = \chi(f^\*\xi) \in H^\*(M'). \tag{30}$$

A consequence of (30) is important as it gives the following. *Theorem 4.2*. If *n* is even, then

$$\chi\left(\check{Q}''(\mathbb{R}^N)\right) \neq 0, \qquad N > n. \tag{31}$$

*Proof*: Since *S<sup>n</sup>* ⊂ ℝ*<sup>N</sup>* for *N* >*n*, we have a bundle map ~*f*, *f* : *TS<sup>n</sup>* ! *<sup>E</sup>* <sup>~</sup>*<sup>ζ</sup> <sup>n</sup>* ℝ*<sup>N</sup>* so

$$\chi(T\mathbb{S}^n) = f^\*\chi\left(\tilde{\zeta}^n(\mathbb{R}^N)\right). \tag{32}$$

However, it is known to be the case that *<sup>χ</sup> TSn* ð Þ is *<sup>χ</sup> <sup>S</sup><sup>n</sup>* ð Þ times the fundamental class of *Sn* and *<sup>χ</sup> Sn* ð Þ¼ <sup>2</sup> 6¼ 0. □

#### **5. A unique one-form constructed from the curvature**

This is an important characteristic class which is important and plays a significant role. Consider principal bundles associated with a smooth oriented *n*-dimensional vector bundle *ξ* over a smooth manifold *M*. There is the principal bundle of frames *F*ð Þ*ξ* of *E*. If *ξ* has a Riemannian metric ,h i, the bundle *O E*ð Þ of orthonormal frames can be considered, which is a principal bundle with group *O n*ð Þ. Since only paracompact *M* are considered here, there is an Ehresmann connection *ω* on *O E*ð Þ. Thus *ω* is a matrix of one-forms *ω<sup>i</sup> j* on *O E*ð Þ with values in *o n*ð Þ, the curvature form <sup>Ω</sup> <sup>¼</sup> *<sup>D</sup><sup>ω</sup>* is a matrix of two-forms Ω*<sup>i</sup> j* � � with values in *o n*ð Þ. A connection *<sup>ω</sup>* on *O E*ð Þ is equivalent to a covariant derivative operator on *E* compatible with the metric, and for a general *ξ* over *M*, there will be many connections compatible with the metric. One can not be singled out by requiring a symmetric connection which only makes sense for the tangent bundle. As *ξ* is oriented, we can also consider the bundle *SO E*ð Þ of positively oriented frames. If *X* is connected, it is simply one of the two components of *O E*ð Þ, with group *SO n*ð Þ and Lie algebra *o n*ð Þ. A connection *ω* on *SO n*ð Þ again has values in *o n*ð Þ as does the matrix of two-forms Ω.

If we specialize to the case of a smooth oriented *n*-dimensional vector bundle *ξ* ¼ *π* : *E* ! *M* over *M*, with *n* ¼ 2*m* even. If ,h i is a Riemannian metric for *ξ* and *ω* is a connection on the corresponding principal bundle *ω* : *SO E*ð Þ! *M*, consider the *n*form which is defined on *SO E*ð Þ

$$2m \cdot m! \operatorname{Pf}(\mathfrak{Q}) = \sum\_{i\_1, \dots, i\_n} \epsilon^{i\_1, \dots, i\_n} \mathfrak{Q}\_{i\_2}^{i\_1} \wedge \dots \wedge \mathfrak{Q}\_{i\_n}^{i\_{n-1}}.\tag{33}$$

The following is an invariant formulation of a previous theorem. Theorem 5.1. There is a unique *n*-form Λ on *M* such that

$$\overline{\boldsymbol{w}}^{\*}\left(\Lambda\right) = \sum\_{i\_1,\cdots,i\_n} \varepsilon^{i\_1,\cdots,i\_n} \Omega\_{i\_2}^{i\_1} \wedge \cdots \wedge \Omega\_{i\_n}^{i\_{n-1}} = \mathcal{Z}^m m! \operatorname{Pf}(\Omega). \tag{34}$$

*Proof*: Let *X*1, ⋯, *Xn* ∈ *Mp Y*1, ⋯, *Yn* ∈*SO E*ð Þ*<sup>u</sup>* be tangent vectors such that *πYi* ! *Xi*, and choose some *<sup>u</sup>* <sup>∈</sup>*ω*�<sup>1</sup> ð Þ *<sup>p</sup>* . Then form <sup>Λ</sup> must satisfy

$$\Lambda(X\_1, \cdots, X\_n) = 2^m \cdot m! \operatorname{Pf}(\Omega)(Y\_1, \cdots, Y\_n) \tag{35}$$

This suffices to give uniqueness. If it can be shown this Λ in (35) is well-defined, then existence can be established.

Suppose different tangent vectors *Z*1, ⋯, *Zn* are taken such that *ωxZi* ¼ *Xi*. Since *ωx*ð Þ¼ *Yi* � *Zi* 0, all *Yi* � *Zi* are vertical. However, Ωð Þ¼ *Y*, *Z* 0 if either *Y* or *Z* is vertical. Consequently,

$$\begin{split} \text{Pf}(Y\_1, \ldots, Y\_n) &= \text{Pf}(\mathfrak{Q})(Z\_1, Y\_2, \ldots, Y\_n) = \text{Pf}(\mathfrak{Q})(Z\_1, Z\_2, Y\_3, \ldots, Y\_n) \\ &= \text{Pf}(\mathfrak{Q})(Z\_1, \ldots, Z\_n) . \end{split} \tag{36}$$

This means the definition of Λ does not depend on the *Yi* selected. Suppose a different *<sup>u</sup>* <sup>∈</sup>*ω*�<sup>1</sup>ð Þ *<sup>p</sup>* is chosen. Then *<sup>u</sup>* <sup>¼</sup> *RA*ð Þ¼ *<sup>u</sup> <sup>u</sup>* � *<sup>A</sup>* for some *<sup>A</sup>* <sup>∈</sup>*SO n*ð Þ, and so let *Yi* ∈*SO E*ð Þ*<sup>π</sup>* be given by *Yi* ¼ *RA* <sup>∗</sup> *Yi* and

$$\begin{split} \text{Pf}(\mathfrak{Q}) \left( \overline{Y}\_{1}, \cdots, \overline{Y}\_{n} \right) &= \text{Pf}(\mathfrak{Q}) (R\_{A\*} \, Y\_{1}, \cdots, R\_{A\*} \, Y\_{n}) = \text{Pf} \left( R\_{A}^{\*} \, \mathfrak{Q} \right) (Y\_{1}, \cdots, Y\_{n}) \\ &= \text{Pf} \left( A^{-1} \mathfrak{Q} A \right) (Y\_{1}, \cdots, Y\_{n}) = \text{Pf}(\mathfrak{Q}) (Y\_{1}, \cdots, Y\_{n}) . \end{split} \tag{37}$$

□

Theorem 5.2. The unique *n*-form Λ in (35) is closed, *d*Λ ¼ 0.

*Proof*: Suppose *<sup>X</sup>*1, <sup>⋯</sup>,*Xn*þ<sup>1</sup> <sup>∈</sup> *Mp* be given and choose *<sup>u</sup>*∈*ω*�<sup>1</sup>*p*<sup>Þ</sup> and *Y*1, ⋯, *Yn*þ<sup>1</sup> ∈ *SO E*ð Þ*<sup>u</sup>* with *ωxYi* ¼ *Xi* and *hYi* the horizontal component of *Yi*. Then working out *d*Λ

$$\begin{split} d\Lambda(X\_1, \ldots, X\_{n+1}) &= d\Lambda(\overline{\alpha}\_{\mathbf{x}} Y\_1, \ldots, \overline{\alpha}\_{\mathbf{x}} Y\_{n+1}) = d\Lambda(\overline{\alpha}\_{\mathbf{x}} hY\_1, \ldots, \overline{\alpha}\_{\mathbf{x}} hY\_{n+1}) \\ &= (\overline{\alpha}^\* \, d\Lambda)(hY\_1, \ldots, hY\_n) = d(\overline{\alpha}^\* \, \Lambda)(hY\_1, \ldots, hY\_{n+1}) \\ &= 2^{\mathfrak{m}} \cdot m! d(\operatorname{Pf}(\Omega))(hY\_1, \ldots, hY\_{n+1}) = 2^{\mathfrak{m}} \cdot m! \mathrm{D}(\operatorname{Pf}(\Omega))(Y\_1, \ldots, Y\_{n+1}). \end{split} \tag{38}$$

However, *D*Ω ¼ 0 by Bianchi's identity and a consequence of this is that (38) vanishes. □

This result applies automatically when *ξ* is the tangent bundle. The implication of this is that the *<sup>n</sup>*-form <sup>Λ</sup> determines a de Rham cohomolgy class ½ � <sup>Λ</sup> <sup>∈</sup> *<sup>H</sup>n*ð Þ *<sup>M</sup>* of *<sup>M</sup>*. The form Λ itself depends on the oriented *n*-dimensional bundle *ξ* ¼ *π* : *E* ! *M* over *M* as well as the choice of metric for *ξ* and connection *ω* on the corresponding bundle *SO E*ð Þ.

Theorem 5.3. The cohomology class ½ � Λ is independent of both the metric and the connection *ω*.

*Proof*: Suppose two metrics ,h i, ,h i<sup>0</sup> are given for *ξ*. Then the corresponding principal bundles *SO E*ð Þ and *SO*<sup>0</sup> ð Þ *<sup>E</sup>* are equivalent. If <sup>~</sup>*<sup>f</sup>* : *SO*<sup>0</sup> ð Þ! *E SO E*ð Þ is a fiber preserving diffeomorphism which commutes with the action *SO n*ð Þ and *ω* a connection on *SO E*ð Þ. Then *<sup>ω</sup>*<sup>0</sup> <sup>¼</sup> <sup>~</sup>*<sup>f</sup>* ∗ *ω* is a connection on *SO E*ð Þ. Corresponding curvature forms satisfy <sup>Ω</sup><sup>0</sup> <sup>¼</sup> <sup>~</sup>*<sup>f</sup>* ∗ Pfð Þ <sup>Ω</sup> so Pf <sup>Ω</sup><sup>0</sup> ð Þ¼ <sup>~</sup>*<sup>f</sup>* ∗ Pfð Þ Ω . The corresponding forms Λ and Λ<sup>0</sup> are in fact equal. It suffices to show any two connection differential forms *ω*0, *ω*<sup>1</sup> on the same *SO E*ð Þ generate forms Λ0, Λ<sup>1</sup> whose difference is exact. If *π* : *M* � ½ �! 0, 1 *M* is the projection *<sup>π</sup>*ð Þ¼ *<sup>p</sup>*, *<sup>t</sup> <sup>p</sup>*, consider the bundle *<sup>π</sup>* <sup>∗</sup> *SO*ð Þ*<sup>ξ</sup>* over *<sup>M</sup>* � ½ � 0, 1 . Induced connections are *<sup>π</sup>* <sup>∗</sup>*ω*<sup>0</sup> and *<sup>π</sup>* <sup>∗</sup> *<sup>ω</sup>*<sup>1</sup> on *<sup>π</sup>* <sup>∗</sup> *SO*ð Þ*<sup>ξ</sup>* . Let *<sup>τ</sup>* : *<sup>M</sup>* � ½ �! 0, 1 ½ � 0, 1 defined here as *τ*ð Þ¼ *p*, *t t* and define a connection

$$
\rho = (\mathbf{1} - \boldsymbol{\tau})\boldsymbol{\pi}^\* \ o\_0 + \boldsymbol{\tau}\boldsymbol{\pi}^\* \ o\_1 \tag{39}
$$

on *<sup>π</sup>* <sup>∗</sup> *SO*ð Þ*<sup>ξ</sup>* with <sup>Ω</sup> the connection form. If *it* maps *<sup>M</sup>* to *<sup>M</sup>* � ½ � 0, 1 and is defined as *it*ð Þ¼ *<sup>p</sup>* ð Þ *<sup>p</sup>*, *<sup>t</sup>* , then *<sup>i</sup>* <sup>∗</sup> <sup>0</sup> ð Þ *<sup>ω</sup>* can be identified with *<sup>ω</sup>*<sup>0</sup> and *<sup>i</sup>* <sup>∗</sup> <sup>1</sup> ð Þ *ω* with *ω*1. By Theorems 5.1 and 5.2, which hold for manifolds with and without boundary, there is a closed *n*-form <sup>Λ</sup> on *<sup>M</sup>* � ½ � 0, 1 which pulls back to 2*<sup>m</sup> <sup>m</sup>*! Pfð Þ <sup>Ω</sup> on the total space of *<sup>π</sup>* <sup>∗</sup> *SO*ð Þ*<sup>ξ</sup>* . A theorem states for any *<sup>k</sup>*-form *<sup>ω</sup>* on *<sup>M</sup>* � ½ � 0, 1 , *<sup>i</sup>* <sup>∗</sup> <sup>1</sup> *<sup>ω</sup>* � *<sup>i</sup>* <sup>∗</sup> <sup>0</sup> *ω* ¼ *d I*ð Þ� *ω I d*ð Þ *ω* . So if *<sup>d</sup><sup>ω</sup>* <sup>¼</sup> 0, this implies *<sup>i</sup>* <sup>∗</sup> <sup>1</sup> *<sup>ω</sup>* � *<sup>i</sup>* <sup>∗</sup> <sup>0</sup> *ω* ¼ *d I*ð Þ *ω* . Substituting the form Λ in place of *ω* into this, it follows that <sup>Λ</sup><sup>1</sup> � <sup>Λ</sup><sup>0</sup> is exact. □

Thus every smooth oriented smaooth bundle *ξ* over *M* of even fibre dimension *n* determines a de Rham cohomology class *<sup>C</sup>*ð Þ¼ *<sup>ξ</sup>* ½ � <sup>Λ</sup> <sup>∈</sup> *<sup>H</sup><sup>n</sup>*ð Þ *<sup>M</sup>* and *<sup>C</sup>*ð Þ¼ *<sup>ξ</sup> <sup>C</sup>*ð Þ*<sup>η</sup>* if *<sup>ξ</sup>*≃*η*. It may be asked how does the object *<sup>C</sup>*ð Þ*<sup>ξ</sup>* behave with respect to *<sup>f</sup>* <sup>∗</sup> .

Theorem 5.4. Let *ξ* ¼ *π* : *ξ* ! *M* be a smooth oriented bundle over *M* with fibre dimension *n* even, let *f* : *M*<sup>0</sup> ! *M* be a smooth map. Then

$$\mathcal{C}(f^\*\xi) = f^\*\,^\*\mathcal{C}(\xi) \in H^n(M'). \tag{40}$$

*Proof*: The total space of *f* <sup>∗</sup> *ξ* is called *E*<sup>0</sup> . Let <sup>~</sup>*<sup>f</sup>* : *<sup>E</sup>*<sup>0</sup> ! *<sup>E</sup>* be the bundle map covering *<sup>f</sup>*. If ,h i is a metric on *<sup>E</sup>*, then <sup>~</sup>*<sup>f</sup>* ∗ h i, is a metric on *E*. There is an equivalence *f* : *SO E*<sup>0</sup> ð Þ! *SO E*ð Þ covering *f* with *ω*<sup>0</sup> taking *SO E*<sup>0</sup> ð Þ to *M*<sup>0</sup> and *ω* mapping *SO E*ð Þ to *M*.

If *ω* is a connection on *SO E*ð Þ, then *f* ∗ ð Þ *ω* is a connection on *SO E*<sup>0</sup> ð Þ. It is seen that the corresponding connection forms satisfy <sup>Ω</sup><sup>0</sup> <sup>¼</sup> <sup>~</sup>*<sup>f</sup>* ∗ Ω. Aa a result, we have

$$\text{Pf}(\Omega') = \text{Pf}\left(\overline{f}^\*\Omega\right) = \overline{f}^\* \text{ Pf}(\Omega). \tag{41}$$

For *n*-forms Λ on *M* given by Theorem 5.1, we then have

$$
\overline{\boldsymbol{\sigma}}^{\prime \*} (\boldsymbol{f}^\* \boldsymbol{\Lambda}) = \overline{\boldsymbol{f}}^\* \, \overline{\boldsymbol{\sigma}}^\* \boldsymbol{\Lambda} = \mathbf{2}^m \cdot \boldsymbol{m}! \overline{\boldsymbol{f}}^\* \, \operatorname{Pf}(\boldsymbol{\Omega}) = \mathbf{2}^m \cdot \boldsymbol{m}! \operatorname{Pf}(\boldsymbol{\Omega}').\tag{42}
$$

This means *<sup>f</sup>* <sup>∗</sup><sup>Λ</sup> must be the *<sup>n</sup>*-form <sup>Λ</sup><sup>0</sup> on *<sup>M</sup>*<sup>0</sup> given in (31). □

When *ξ* is a smooth oriented bundle of odd fibre dimension, the definition of *C* may be extended. It would be remarkable if it were the case that *C*ð Þ*ξ* were always a constant multiple of *χ ξ*ð Þ. To this end, the following theorem is needed.

Theorem 5.5. Let *ξ<sup>i</sup>* ¼ *π<sup>i</sup>* : *Ei* ! *M* for *i* ¼ 1, 2 be smooth oriented vector bundles over *M* of fibre dimension *n*<sup>1</sup> and *n*2. If *ni* ¼ 2*mi*, then

$$\mathcal{C}(\xi\_1 \oplus \xi\_2) = \frac{(m\_1 + m\_2)!}{m\_1! m\_2!} \mathcal{C}(\xi\_1) \cup \mathcal{C}(\xi\_2). \tag{43}$$

If *n*<sup>1</sup> or *n*<sup>2</sup> is odd, this reduces to *C ξ*1⊕*ξ*<sup>2</sup> ð Þ¼ 0.

*Proof*: Pick two metrics which are Riemannian for each *ξ<sup>i</sup>* and set h i �, � ¼ h i �, � <sup>1</sup>⊕h i �, � <sup>2</sup> on *ξ*1⊕*ξ*<sup>2</sup> ¼ *π* : *E* ! *M*. Let *ω<sup>i</sup>* : *SO E*ð Þ!*<sup>i</sup> M* and *ω* : *SO E*ð Þ! *M* be the corresponding principal bundles. Over *M* consider the product principal bundle *Q* ¼ *SO E*ð Þ<sup>1</sup> ∗ *SO E*ð Þ<sup>2</sup> with corresponding group *SO n*ð Þ� <sup>1</sup> *SO n*ð Þ<sup>2</sup> ⊂*SO n*ð Þ <sup>1</sup> þ *n*<sup>2</sup> whose fiber over *p* ∈ *M* is the direct product *ω*�<sup>1</sup> <sup>1</sup> � *<sup>ω</sup>*�<sup>1</sup> <sup>2</sup> ð Þ *p* , so this bundle is a subset of *SO E*ð Þ.

Let *ρ<sup>i</sup>* be the projection maps for *Q* which project this down onto either of its factors. If *ω<sup>i</sup>* are connections on *SO E*ð Þ*<sup>i</sup>* , with curvature forms Ω*i*, then

$$
\rho\_1^\* \, a\_1 \oplus \rho\_2^\* \, a\_2 = \begin{pmatrix} \rho\_1^\* \, a\_1 & \mathbf{0} \\ \mathbf{0} & \rho\_2^\* \, a\_2 \end{pmatrix} \tag{44}
$$

is a connection on *Q* and the curvature form is

$$
\boldsymbol{\Omega} = \rho\_1^\* \,\boldsymbol{\Omega}\_1 \oplus \rho\_2^\* \,\boldsymbol{\Omega}\_2 = \begin{pmatrix} \rho\_1^\* \,\boldsymbol{\Omega}\_1 & \mathbf{0} \\ \mathbf{0} & \rho\_2^\* \,\boldsymbol{\Omega}\_2 \end{pmatrix} \tag{45}
$$

The connection *ω* can be extended uniquely to a connection *ω*~ on *SO E*ð Þ. The requirement *ω σ* ~ð Þ¼ ð Þ *M M* determines *ω*~ at the new vertical vectors, hence *ω*~ is determined at all points of *Q*, and then at all points of *SO E*ð Þ by the requirement *ω*~ *R*<sup>∗</sup> *<sup>A</sup> <sup>Y</sup>* <sup>¼</sup> *Ad A*�<sup>1</sup> *<sup>ω</sup>*~ð Þ *<sup>Y</sup>* .

At any point *e*∈ *Q*, the horizontal vectors for *ω*~ are the same as that for *ω*. At *E*, it holds that <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup><sup>~</sup> for tangent vectors to *<sup>Q</sup>* which implies, using Pfð Þ¼ *<sup>A</sup>*⊕*<sup>B</sup>* Pfð Þ� *A* Pfð Þ *B* , that

$$\operatorname{Pf}(\tilde{\Omega}) = \operatorname{Pf}(\overline{\Omega}) = \operatorname{Pf}(\rho\_1^\* \Omega\_1) \wedge \operatorname{Pf}(\rho\_2^\* \Omega\_2) = \rho\_1^\* \operatorname{Pf}(\Omega\_1) \wedge \rho\_2^\* \ (\Omega\_2). \tag{46}$$

Consequently, if Λ*<sup>i</sup>* are the forms given by (34), then at *e* it must hold that on tangent vectors to *Q*

$$\begin{split} \overline{\boldsymbol{\alpha}}^{\*} \boldsymbol{\Lambda} &= 2^{m\_{1}+m\_{2}} (\boldsymbol{m}\_{1}+\boldsymbol{m}\_{2})! \operatorname{Pf} (\boldsymbol{\tilde{\Omega}}) = \frac{(\boldsymbol{m}\_{1}+\boldsymbol{m}\_{2})!}{m\_{1}! m\_{2}!} 2^{m\_{1}} m\_{1}! \boldsymbol{\rho}\_{1}^{\*} \operatorname{Pf} (\boldsymbol{\Omega}\_{1}) \wedge 2^{m\_{2}} m\_{2}! \boldsymbol{\rho}\_{2}^{\*} \operatorname{Pf} (\boldsymbol{\Omega}\_{2}) \\ &= \frac{(\boldsymbol{m}\_{1}+\boldsymbol{m}\_{2})!}{m\_{1}! m\_{2}!} \boldsymbol{\rho}\_{1}^{\*} \overline{\boldsymbol{\sigma}}\_{1}^{\*} \boldsymbol{\Lambda}\_{1} \wedge \boldsymbol{\rho}\_{2}^{\*} \overline{\boldsymbol{\sigma}}\_{2}^{\*} \boldsymbol{\Lambda}\_{2} = \frac{(\boldsymbol{m}\_{1}+\boldsymbol{m}\_{2})!}{m\_{1}! m\_{2}!} \overline{\boldsymbol{\sigma}}\_{1}^{\*} \boldsymbol{\Lambda}\_{1} \wedge \overline{\boldsymbol{\sigma}}\_{2}^{\*} \boldsymbol{\Lambda}\_{2} . \end{split} \tag{47}$$

This implies that

$$
\Lambda = \frac{(m\_1 + m\_2)!}{m\_1! m\_2!} \Lambda\_1 \wedge \Lambda\_2. \tag{48}
$$

$$
\Box
$$

Corollary 5.1. If the oriented bundle *ξ* ¼ *π* : *E* ! *M* has a nowhere zero section *s*, then

$$\mathbf{C}(\xi) = \mathbf{0}.\tag{49}$$

*Proof:* Let *E*<sup>1</sup> ⊂*E* be written

$$\cup\_{p} \cup\_{M} \mathbb{R} \cdot \mathfrak{s}(p),\tag{50}$$

and let *E*<sup>2</sup> ⊂*E* be the orthogonal complement

$$\cup\_{p} \mathbb{C}\_{\mathbb{R}}(\mathbb{R} \cdot s(p))^{\perp} \tag{51}$$

with respect to some Riemannian metric on *E*. Then *ξ*<sup>1</sup> ¼ *π*1∣*E*<sup>1</sup> : *E*<sup>1</sup> ! *M* is an oriented one-dimensional bundle. Consequently, *ξ*<sup>2</sup> ¼ *π*2∣*E*<sup>2</sup> : *E*<sup>2</sup> ! *M* is also an oriented bundle since *ξ* is oriented. Clearly *ξ*≃*ξ*1⊕*ξ*2. An application of the previous result shows that *<sup>C</sup>*ð Þ¼ *<sup>ξ</sup>* 0. □

This theorem is almost enough to characterize *χ* as we can now show the statement which relates *C*ð Þ*ξ* and the Euler class.

Corollary 5.2. If *ξ* ¼ *π* : *E* ! *M* is a smooth vector bundle of fibre dimension *n* over a compact oriented manifold *<sup>M</sup>*, then the class *<sup>C</sup>*ð Þ*<sup>ξ</sup>* <sup>∈</sup> *<sup>H</sup><sup>n</sup>*ð Þ *<sup>M</sup>* is a multiple of the Euler class *χ ξ*ð Þ.

*Proof*: Suppose *S* is the sphere bundle *S* ¼ f g *e*∈*E* : h i *e*,*e* ¼ 1 , which is constructed with respect to some Riemannian metric on *E*. Let *π*<sup>0</sup> : *S* ! *X* be the restriction *π*∣*S*. The bundle *π* <sup>∗</sup> <sup>0</sup> *ξ* has a nowhere zero section. Corrollary 5.1 and Theorem 5.4 then yield

$$
\pi\_0^\* \mathcal{C}(\xi) = \mathcal{C}(\pi\_0^\* \xi) = 0. \tag{52}
$$

However, there is a theorem which states a class *<sup>α</sup>* <sup>∈</sup> *<sup>H</sup><sup>n</sup>*ð Þ *<sup>M</sup>* satisfies *<sup>π</sup>* <sup>∗</sup> <sup>0</sup> *α* ¼ 0 if and only if *α* is a multiple of *χ ξ*ð Þ. It can now be inferred that *C*ð Þ*ξ* is a multiple of the Euler class *χ ξ*ð Þ. □

#### **6. The Gauss-Bonnet-Chern theorem**

If Corollary 5.2 is applied to the tangent bundle of a compact oriented manifold *M* of dimension *<sup>n</sup>* which is even, the class *C TM*ð Þ<sup>∈</sup> *<sup>H</sup><sup>n</sup>*ð Þ *<sup>M</sup>* is some multiple of the Euler class *<sup>χ</sup>*ð Þ *TM* . This fact is not so interesting because *<sup>H</sup><sup>n</sup>*ð Þ *<sup>M</sup>* is one-dimensional since it means *C TM*ð Þ¼ 0 if *χ*ð Þ¼ *TM* 0. The corollary does lead to something interesting when applied to the universal bundle.

Theorem 6.1 For every even *n*, there is a constant *β<sup>n</sup>* such that

$$\mathbf{C}(\xi) = \beta\_n \chi(\xi). \tag{53}$$

*An Introduction to the Generalized Gauss-Bonnet-Chern Theorem DOI: http://dx.doi.org/10.5772/intechopen.105716*

for all smooth oriented *n*-dimensional bundles *ξ* over compact oriented manifolds. In this sense, it is universal.

*Proof*: Begin with the bundles ~*zetan* ℝ*<sup>N</sup>* � � for *N* >*n*. Corollary 5.2 implies there are constants *βn*,*<sup>N</sup>* such that

$$\mathbf{C}\left(\check{\boldsymbol{\zeta}}^{\boldsymbol{n}}\left(\mathbb{R}^{N}\right)\right) = \boldsymbol{\beta}\_{\boldsymbol{n},N}\boldsymbol{\chi}\left(\check{\boldsymbol{\zeta}}^{\boldsymbol{n}}\left(\mathbb{R}^{N}\right)\right) \in H^{\boldsymbol{n}}\left(\check{\boldsymbol{\zeta}}\_{\boldsymbol{n}}\left(\mathbb{R}^{N}\right)\right).\tag{54}$$

If *<sup>j</sup>* : *<sup>G</sup>*<sup>~</sup> *<sup>n</sup>* <sup>ℝ</sup>*<sup>N</sup>* � � ! *<sup>G</sup>*<sup>~</sup> *<sup>n</sup>* <sup>ℝ</sup>*<sup>M</sup>* � � is the natural inclusion, then *<sup>j</sup>* <sup>∗</sup> ~*ζ <sup>N</sup>* ℝ*<sup>N</sup>* � � � � <sup>≃</sup>~*<sup>ζ</sup> <sup>n</sup>* ℝ*<sup>N</sup>* � �. Equation (30) and Theorem 5.4 yield

$$\mathbf{C}\left(\tilde{\boldsymbol{\zeta}}''\left(\mathbb{R}^N\right)\right) = j^\* \, \mathbf{C}\left(\tilde{\boldsymbol{\zeta}}''\left(\mathbb{R}^M\right)\right) = j^\* \, \mathbf{x}\left(\tilde{\boldsymbol{\zeta}}''\left(\mathbb{R}^M\right)\right). \tag{55}$$

Thus, (54), (55) give

$$
\beta\_{n,N} \chi \left( \tilde{\boldsymbol{\zeta}}^n \left( \mathbb{R}^N \right) \right) = \beta\_{n,M} \chi \left( \tilde{\boldsymbol{\zeta}}^n \left( \mathbb{R}^N \right) \right). \tag{56}
$$

Since *χ* ~*ζ <sup>n</sup>* ℝ*<sup>N</sup>* � � � � 6¼ 0 by Theorem 4.2, this implies that *<sup>β</sup><sup>n</sup>*,*<sup>N</sup>* <sup>¼</sup> *<sup>β</sup><sup>n</sup>*,*<sup>M</sup>* for all *<sup>M</sup>*, *<sup>N</sup>* <sup>&</sup>gt;1. This common number is called *βn*, and we have

$$\mathcal{C}\left(\tilde{\zeta}^{\mathfrak{n}}\left(\mathbb{R}^{N}\right)\right) = \beta\_{\mathfrak{n}}\chi\left(\tilde{\zeta}^{\mathfrak{n}}\left(\mathbb{R}^{N}\right)\right). \tag{57}$$

However it is known that any smooth oriented *n*-dimensional bundle *ξ* over a compact manifold *M* is equivalent to *f* <sup>∗</sup> ~*ζ <sup>n</sup>* ℝ*<sup>N</sup>* � � � � for some smooth map *<sup>f</sup>* : *<sup>M</sup>* ! *G*~ ℝ*<sup>N</sup>* � �, then

$$\mathbf{C}(\boldsymbol{\xi}) = \mathbf{C}\left(\boldsymbol{f}^\* \boldsymbol{\bar{\zeta}}^n(\mathbb{R}^N)\right) = \boldsymbol{f}^\* \cdot \mathbf{C}\left(\boldsymbol{\bar{\zeta}}^n(\mathbb{R}^N)\right) = \beta\_n \boldsymbol{f}^\* \cdot \boldsymbol{\chi}\left(\boldsymbol{\bar{\zeta}}^n(\mathbb{R}^N)\right) = \beta\_n \boldsymbol{\chi}(\boldsymbol{\xi}).\tag{58}$$

The constant *β<sup>N</sup>* is universal in nature and it may be asked whether it can be computed. Since it has this universality property, it suffices to compute this constant for a special case where the calculation is easier and in turn implies another application of the next theorem.

Theorem 6.2 For integer *n* ¼ 2*m*, the constant *β<sup>n</sup>* in Theorem 6.1 is

$$\beta\_n = \frac{n!}{2} V(\mathbb{S}^n) = \pi^m 2^n m! \,. \tag{59}$$

If ð Þ *M*, ,h i is a compact manifold of even dimension *n* ¼ 2*m* then

$$\int\_{M} K\_{n} \, d\mu\_{\rm g} = \frac{\pi^{m} 2^{n} m!}{n!} \, \chi(M). \tag{60}$$

*Proof*: Let *ξ* be the tangent bundle *TM* of a compact oriented manifold of dimension *n*. Now (17) gives a formula for *Kn dμg*, where Ω*<sup>i</sup> <sup>j</sup>* are curvature forms for some

□

positively oriented orthonormal moving frame. This implies that the form Λ in Theorem 5.1 for the bundle *SO*ð Þ¼ *ξ SO TM*ð Þ is

$$
\Lambda = n! K\_n d\mu\_g. \tag{61}
$$

If *κ* is the fundamental class of *M* then

$$\chi\left(\int\_{M} K\_n d\mu\_{\underline{g}}\right)\kappa = \frac{1}{n!} \left(\int\_{M} \Lambda\right)\kappa = \frac{1}{n!} \mathbf{C}(\boldsymbol{\xi}) = \frac{\boldsymbol{\beta}\_n}{n!} \boldsymbol{\chi}(\boldsymbol{\xi}) = \frac{\boldsymbol{\beta}\_n}{n!} \boldsymbol{\chi}(M) \cdot \kappa. \tag{62}$$

Hence, equating the coefficients of *κ* on both sides,

$$\int\_{M} K\_{n} d\mu\_{\rm g} = \frac{\beta\_{n}}{n!} \chi(\mathcal{M}).\tag{63}$$

Consider the case of a specific manifold *<sup>M</sup>* <sup>¼</sup> *<sup>S</sup><sup>n</sup>* in (63), where *Kn* <sup>¼</sup> 1 so the left side of (63) reduces to *V Sn* ð Þ

$$V(\mathbf{S}^n) = \frac{\beta\_n}{n!} \chi(\mathbf{S}^n) = \frac{2\beta\_n}{n!}.\tag{64}$$

Since the volume *V S<sup>n</sup>* ð Þ is known to be *<sup>π</sup><sup>m</sup>* <sup>2</sup>*<sup>n</sup>*þ<sup>1</sup> *<sup>m</sup>*!*=n*!, solve (64) for *<sup>β</sup><sup>n</sup>* in terms of *V S<sup>n</sup>* ð Þ ,

$$\beta\_n = \frac{n!}{2} V(\mathbf{S}^n) = \pi^n \mathbf{2}^n m! . \tag{65}$$

This value of *β<sup>n</sup>* can be put back into (60) and for the manifold *M*, it follows that

$$\int\_{M} K\_{n} \, d\mu\_{\rm g} = \frac{\pi^{m} \, 2^{n} m!}{n!} \, \chi(M). \tag{66}$$

□

#### **7. The theorem for manifolds with boundary**

Theorem 5.5 played a large part in the proof of (64). It allowed us to state that if *ξ* ¼ *π* : *E* ! *M* is an oriented *n*-dimensional vector bundle with sphere bundle *π*<sup>0</sup> : *<sup>S</sup>*<sup>0</sup> ! *<sup>M</sup>*, then *<sup>π</sup>* <sup>∗</sup> <sup>0</sup> *C*ð Þ¼ *ξ* 0 was a large part of the proof of (). If Λ is the *n*-form on *M* representing *<sup>C</sup>*ð Þ*<sup>ξ</sup>* , then the *<sup>n</sup>*-form *<sup>π</sup>* <sup>∗</sup> <sup>0</sup> Λ on *S* is exact *π* <sup>∗</sup> <sup>0</sup> Λ ¼ *d*Φ for some ð Þ *n* � 1 -form on *<sup>S</sup>*. Suppose *<sup>ξ</sup>* <sup>¼</sup> *TM*<sup>∗</sup> and let *<sup>X</sup>* be a unit vector bundle on *<sup>M</sup>* which has an isolated singularity at *p* ∈ *M* Let *B*ð Þ*ε* be a closed ball of radius *ε* around *p* and set *M<sup>ε</sup>* ¼ *M*nintð Þ *B*ð Þ*ε* where int denotes the interior. Then *X M*ð Þ*<sup>ε</sup>* is a manifold with boundary, the image of *M<sup>ε</sup>* under the section *X* : *M*nf g*p* ! *S*. Consequently

$$\int\_{M} \Lambda = \int\_{M - \{p\}} \Lambda = \lim\_{\varepsilon \to 0} \int\_{M\_{\varepsilon}} \Lambda = \lim\_{\varepsilon \to 0} \int\_{M\_{\varepsilon}} X^\* \left(\pi\_0^\* \Lambda\right) = \lim\_{\varepsilon \to 0} \int\_{X(M\_{\varepsilon})} \pi\_0^\* \Lambda$$

$$= \lim\_{\varepsilon \to 0} \int\_{X(M\_{\varepsilon})} d\Phi = \lim\_{\varepsilon \to 0} \int\_{\partial X(M\_{\varepsilon})} \Phi. \tag{67}$$

If indð Þ *X*, *p* is the index of *X* at *p*

$$\int\_{\mathcal{M}} \Lambda = \text{ind}(X, p) \int\_{\pi\_0^{-1}(p)} \Phi = \chi(\mathcal{M}) \int\_{\pi\_0^{-1}(p)} \Phi. \tag{68}$$

Since *n* ¼ 2*m* we also have the Gauss-Bonnet-Chern Theorem 6.2

$$\int\_{M} \Lambda = \int\_{M} n! K\_{n} d\mu\_{\rm g} = \pi^{m} m! 2^{n} \chi(M),\tag{69}$$

we finally obtain

$$\int\_{\pi\_0^{-1}(p)} \Phi = \pi^m m! 2^n. \tag{70}$$

Let ð Þ *<sup>M</sup>*, *<sup>∂</sup><sup>M</sup>* be a compact orientable manifold with boundary with Euler characteristic *<sup>χ</sup>*ð Þ¼ *<sup>M</sup>* dim*H*<sup>0</sup>ð Þ� *<sup>M</sup>* dim*H*<sup>1</sup> ð Þþ *<sup>M</sup>* <sup>⋯</sup>. A compact oriented manifold *<sup>M</sup>*<sup>2</sup> can be constructed, the double of *M*, by taking two disjoint copies of *M* and identifying corresponding points of *∂M*.

Theorem 7.1 The Euler characteristic of the manifold *M*<sup>2</sup> is given by

$$
\chi(\mathsf{M}^2) = \mathfrak{Z}\chi(\mathsf{M}) - \chi(\partial \mathsf{M}).\tag{71}
$$

*Proof*: Let *U* and *V* be open neighborhoods of the two copies of *M* in *M*<sup>2</sup> such that *<sup>H</sup><sup>k</sup>*ð Þ� *<sup>U</sup> H V*ð Þ� *<sup>H</sup><sup>k</sup>* ð Þ *<sup>M</sup>* for all *<sup>k</sup>* and *<sup>H</sup><sup>k</sup>*ð Þ� *<sup>U</sup>*∩*<sup>V</sup> <sup>H</sup><sup>k</sup>*ð Þ *<sup>∂</sup><sup>M</sup>* for all *<sup>k</sup>*. So there is the sequence 0 ! *<sup>H</sup>*<sup>0</sup> *<sup>M</sup>*<sup>2</sup> � � ! <sup>⋯</sup> ! *<sup>H</sup><sup>k</sup> <sup>M</sup>*<sup>2</sup> � � ! *<sup>H</sup><sup>k</sup>*ð Þ *<sup>U</sup>* <sup>⊕</sup>*H<sup>k</sup>* ð Þ! *<sup>V</sup>* <sup>⋯</sup> ! *<sup>H</sup><sup>k</sup>*ð Þ! *<sup>U</sup>*∩*<sup>V</sup>*

*<sup>H</sup><sup>k</sup>*þ<sup>1</sup> *<sup>M</sup>*<sup>2</sup> � �. When the sequence is exact, a theorem can be applied to obtain the result. □ This is very interesting since it claims different things depending on whether the dimension *<sup>n</sup>* of *<sup>M</sup>* is even or odd. When *<sup>n</sup>* is odd *<sup>χ</sup> <sup>M</sup>*<sup>2</sup> � � <sup>¼</sup> 0 hence *<sup>χ</sup>*ð Þ¼ *<sup>M</sup>* <sup>1</sup>*=*2*χ*ð Þ *<sup>∂</sup><sup>M</sup>* which implies *<sup>χ</sup>*ð Þ *<sup>∂</sup><sup>M</sup>* must be even. But when *<sup>n</sup>* is even, *<sup>χ</sup>*ð Þ¼ *<sup>∂</sup><sup>M</sup>* 0, so the previous theorem implies

$$
\mathfrak{Q}\chi(\mathsf{M}) = \chi\left(\mathsf{M}^{2}\right). \tag{72}
$$

Corollary 7.1 Let *M* be a compact orientable manifold with boundary of even dimension *<sup>n</sup>*. Let *<sup>X</sup>* be a vector field on *<sup>M</sup>* with only finitely many zeros all in *<sup>M</sup>*n*∂<sup>M</sup>* such that *<sup>X</sup>* is outward pointing on *<sup>∂</sup>M*. The sum of indices of *<sup>X</sup>* is *<sup>χ</sup>*ð Þ *<sup>M</sup>* .

*Proof*: Modify *X* near *∂M* so it is an outward pointing unit normal *ν* on the boundary and so there are no new zeros. Then there is a vector field on *M*<sup>2</sup> which looks like *X* on one copy of *M* and �*X* on the other. Since *n* is even, the index �*X* of an isolated zero is the same as the index of *X* at that zero. The Theorem of Poincaré-Hopf on the sum of indices of *<sup>X</sup>* gives twice the sum of the indices of *<sup>X</sup>* equals *<sup>χ</sup> <sup>M</sup>*<sup>2</sup> � � <sup>¼</sup> <sup>2</sup>*χ*ð Þ *<sup>M</sup>* by (72). □

Theorem 7.2 Let *M* be a compact oriented Riemannian manifold with boundary of even dimension *n* ¼ 2*m*, tangent bundle *π* : *TM* ! *M* and associated sphere bundle *π*<sup>0</sup> ¼ *π*∣*S* : *S* ! *M*. Let *ω* be a connection on the principal bundle *ω* : *SO TM*ð Þ! *M*, with curvature form Ω. Let Λ be the unique *n*-form on *M* with

$$\overline{\boldsymbol{w}}^{\*}\Lambda = \sum \boldsymbol{\varepsilon}^{i\_{1},\cdots,i\_{n}}\Omega\_{i\_{1}}^{i\_{1}}\wedge\cdots\wedge\Omega\_{i\_{n}}^{i\_{n-1}} = \mathcal{Z}^{m}\boldsymbol{m}!\operatorname{Pf}(\Omega),\tag{73}$$

and <sup>Φ</sup> an ð Þ *<sup>n</sup>* � <sup>1</sup> -form on *<sup>S</sup>* with *<sup>π</sup>* <sup>∗</sup> <sup>0</sup> <sup>Λ</sup> <sup>¼</sup> *<sup>d</sup>*Φ. Let *<sup>ν</sup>* : *<sup>∂</sup><sup>M</sup>* ! *<sup>S</sup>* be the outward pointing unit normal on *∂M*. Then

$$\int\_{M} K\_{n} \, d\mu\_{\rm g} = \frac{1}{n!} \int\_{M} \Lambda = \frac{\pi^{m} m! 2^{n}}{n!} \, \chi(M) + \frac{1}{n!} \int\_{\partial M} \nu^{\*} \, \Phi. \tag{74}$$

*Proof*: Extend *ν* to a vector field *X* on *M* with only finitely many zeros *<sup>p</sup>*1, … , *pk* <sup>∈</sup> *<sup>M</sup>*n*∂M*. Let *Bi*ð Þ*<sup>ε</sup>* be the closed balls of radius *<sup>ε</sup>* centered at *pi* which are disjoint from each other and from *<sup>∂</sup>M*. Put *<sup>M</sup><sup>ε</sup>* <sup>¼</sup> *<sup>M</sup>*n∪*<sup>k</sup> <sup>i</sup>*¼<sup>1</sup> int*Bi*ð Þ*<sup>ε</sup>* . Now integrate the form Λ over *M* and use (70)

$$\int\_{M} \Lambda = \lim\_{\varepsilon \to 0} \int\_{\partial \mathcal{X}(\mathcal{M}\_{\varepsilon})} \Phi = \int\_{\nu(\partial \mathcal{M})} \Phi + \sum\_{i=1}^{k} \lim\_{\varepsilon \to 0} \int\_{\partial \mathcal{B}\_{i}(\varepsilon)} \Phi$$

$$= \int\_{\partial \mathcal{M}} \nu^\* \Phi + \pi^m m! 2^n \sum\_{i=1}^{M} (\operatorname{ind} \mathcal{X})\_{p\_i} = \int\_{\partial \mathcal{M}} \nu^\* \Phi + \pi^m m! 2^n \chi(\mathcal{M}).$$

The last line makes use of Corollary 7.1. □

Theorem 7.2 presents one way in which Theorem 6.2 can be generalized to the case of manifolds with boundary. At this point an interpretation for the first term in (75) is not easy to provide. It is required to obtain an explicit Φ such that *π* <sup>∗</sup> <sup>0</sup> Λ ¼ *d*Φ. In fact such a Φ can be constructed.

#### **Author details**

Paul Bracken Department of Mathematics, University of Texas, Edinburg, TX, USA

\*Address all correspondence to: paul.bracken@utrgv.edu

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*An Introduction to the Generalized Gauss-Bonnet-Chern Theorem DOI: http://dx.doi.org/10.5772/intechopen.105716*

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Section 2
