1. Introduction

Let <sup>i</sup> : <sup>M</sup> ! <sup>M</sup><sup>~</sup> be an immersion of an <sup>n</sup>-dimensional manifold <sup>M</sup> into an <sup>m</sup>-dimensional Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ. Denote by <sup>g</sup> <sup>¼</sup> <sup>i</sup> �~g the induced Riemannian metric on M. Thus, i become an isometric immersion and M is also a Riemannian manifold with the Riemannian metric gðX, YÞ ¼ ~gðX, YÞ for any vector fields X, Y in M. The Riemannian metric g on M is called the induced metric on <sup>M</sup>. In local components, gij <sup>¼</sup> gABB<sup>B</sup> <sup>j</sup> B<sup>A</sup> <sup>i</sup> with <sup>g</sup> <sup>¼</sup> gjidxj dxj and <sup>~</sup><sup>g</sup> <sup>¼</sup> gBAdUBdUA.

If a vector field ξ<sup>p</sup> of M~ at a point p∈M satisfies

$$
\tilde{\mathcal{G}}(X\_p, \xi\_p) = 0 \tag{1}
$$

for any vector Xp of M at p, then ξ<sup>p</sup> is called a normal vector of M in M~ at p. A unit normal vector field of M in M~ is called a normal section on M [3].

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2017 The Author(s). Licensee InTech. 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.

distribution, and eproduction in any medium, provided the original work is properly cited.

By T<sup>⊥</sup>M, we denote the vector bundle of all normal vectors of M in M~ . Then, the tangent bundle of M~ is the direct sum of the tangent bundle TM of M and the normal bundle T<sup>⊥</sup>M of M in M~ , i:e:,

$$T\tilde{M} = TM \oplus T^{\perp}M.\tag{2}$$

We note that if the sub-manifold M is of codimension one in M~ and they are both orientiable, we can always choose a normal section ξ on M, i:e:,

$$g(X,\xi) = 0, \ g(\xi,\xi) = 1,\tag{3}$$

where X is any arbitrary vector field on M.

By ∇~ , denote the Riemannian connection on M~ and we put

$$
\tilde{\nabla}\_X Y = \nabla\_X Y + h(X, Y) \tag{4}
$$

for any vector fields X, Y tangent to M, where ∇XY and hðX, YÞ are tangential and the normal components of <sup>∇</sup><sup>~</sup> XY, respectively. Formula <sup>ð</sup>4<sup>Þ</sup> is called the Gauss formula for the sub-manifold <sup>M</sup> of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ.

Proposition 1.1. ∇ is the Riemannian connection of the induced metric g ¼ i �~g on M and hðX, YÞ is a normal vector field over M, which is symmetric and bilinear in X and Y.

Proof: Let α and β be differentiable functions on M. Then, we have

$$\begin{aligned} \mathring{\nabla}\_{\alpha X}(\beta Y) &= \alpha \{ X(\beta)Y + \beta \mathring{\nabla}\_X Y \} \\ &= \alpha \{ X(\beta)Y + \beta \nabla\_X Y + \beta h(X, Y) \} \end{aligned} $$
 
$$ \nabla\_{\alpha X} \beta Y + h(\alpha X, \beta Y) = \alpha \beta \nabla\_X Y + \alpha X(\beta)Y + \alpha \beta h(X, Y) \tag{5} $$

This implies that

$$
\nabla\_{\alpha X}(\beta Y) = \alpha X(\beta)Y + \alpha \beta \nabla\_X Y \tag{6}
$$

and

$$
\hbar h(\alpha X, \beta Y) = \alpha \beta h(X, Y). \tag{7}
$$

Eq. (6) shows that ∇ defines an affine connection on M and Eq. (4) shows that h is bilinear in X and Y since additivity is trivial [1].

Since the Riemannian connection ∇~ has no torsion, we have

$$0 = \tilde{\nabla} \chi Y - \tilde{\nabla}\_Y X - [X, Y] = \nabla \chi Y + h(X, Y) - \nabla \chi Y - h(Y, X) - [X, Y].\tag{8}$$

By comparing the tangential and normal parts of the last equality, we obtain

$$
\nabla\_X Y - \nabla\_Y X = [X, Y] \tag{9}
$$

and

By T<sup>⊥</sup>M, we denote the vector bundle of all normal vectors of M in M~ . Then, the tangent bundle of M~ is the direct sum of the tangent bundle TM of M and the normal bundle T<sup>⊥</sup>M of

We note that if the sub-manifold M is of codimension one in M~ and they are both orientiable,

for any vector fields X, Y tangent to M, where ∇XY and hðX, YÞ are tangential and the normal components of <sup>∇</sup><sup>~</sup> XY, respectively. Formula <sup>ð</sup>4<sup>Þ</sup> is called the Gauss formula for the sub-manifold

<sup>∇</sup><sup>~</sup> <sup>α</sup><sup>X</sup>ðβYÞ ¼ <sup>α</sup>fXðβÞ<sup>Y</sup> <sup>þ</sup> <sup>β</sup>∇<sup>~</sup> XY<sup>g</sup>

Eq. (6) shows that ∇ defines an affine connection on M and Eq. (4) shows that h is bilinear in X

¼ αfXðβÞY þ β∇XY þ βhðX, YÞg ∇α<sup>X</sup>βY þ hðαX, βYÞ ¼ αβ∇XY þ αXðβÞY þ αβhðX,YÞ (5)

<sup>0</sup> <sup>¼</sup> <sup>∇</sup><sup>~</sup> XY−∇<sup>~</sup> YX−½X, <sup>Y</sup>� ¼ <sup>∇</sup>XY <sup>þ</sup> <sup>h</sup>ðX,YÞ−∇XY−hðY, <sup>X</sup>Þ−½X, <sup>Y</sup>�: (8)

∇α<sup>X</sup>ðβYÞ ¼ αXðβÞY þ αβ∇XY (6)

hðαX, βYÞ ¼ αβhðX, YÞ: (7)

Proposition 1.1. ∇ is the Riemannian connection of the induced metric g ¼ i

Proof: Let α and β be differentiable functions on M. Then, we have

hðX, YÞ is a normal vector field over M, which is symmetric and bilinear in X and Y.

we can always choose a normal section ξ on M, i:e:,

By ∇~ , denote the Riemannian connection on M~ and we put

where X is any arbitrary vector field on M.

<sup>M</sup> of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ.

and Y since additivity is trivial [1].

Since the Riemannian connection ∇~ has no torsion, we have

By comparing the tangential and normal parts of the last equality, we obtain

This implies that

and

TM<sup>~</sup> <sup>¼</sup> TM⊕T<sup>⊥</sup>M: (2)

gðX, ξÞ ¼ 0, gðξ, ξÞ ¼ 1, (3)

<sup>∇</sup><sup>~</sup> XY <sup>¼</sup> <sup>∇</sup>XY <sup>þ</sup> <sup>h</sup>ðX,Y<sup>Þ</sup> (4)

�~g on M and

M in M~ , i:e:,

48 Manifolds - Current Research Areas

$$h(X,Y) = h(Y,X). \tag{10}$$

These equations show that ∇ has no torsion and h is a symmetric bilinear map. Since the metric ~g is parallel, we can easily see that

$$\begin{aligned} (\nabla\_X \mathbf{g})(Y, Z) &= (\tilde{\nabla}\_X \tilde{\mathbf{g}})(Y, Z) \\ &= \tilde{\mathbf{g}}(\tilde{\nabla}\_X Y, Z) + \tilde{\mathbf{g}}(Y, \tilde{\nabla}\_X Z) \\ &= \tilde{\mathbf{g}}\left(\nabla\_X Y + h(X, Y), Z\right) + \tilde{\mathbf{g}}(Y, \nabla\_X Z + h(X, Z)) \\ &= \tilde{\mathbf{g}}\left(\nabla\_X Y, Z\right) + \tilde{\mathbf{g}}(Y, \nabla\_X Z) \\ &= \mathbf{g}(\nabla\_X Y, Z) + \mathbf{g}(Y, \nabla\_X Z) \end{aligned} \tag{11}$$

for any vector fields X, Y, Z tangent to M, that is, ∇ is also the Riemannian connection of the induced metric g on M.

We recall h the second fundamental form of the sub-manifold M (or immersion i), which is defined by

$$h: \Gamma(TM) \times \Gamma(TM) \to \Gamma(T^{\perp}M). \tag{12}$$

If <sup>h</sup> <sup>¼</sup> 0 identically, then sub-manifold <sup>M</sup> is said to be totally geodesic, where <sup>Γ</sup>ðT<sup>⊥</sup>M<sup>Þ</sup> is the set of the differentiable vector fields on normal bundle of M.

Totally geodesic sub-manifolds are simplest sub-manifolds.

Definition 1.1. Let M be an n-dimensional sub-manifold of an m-dimensional Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ. By <sup>h</sup>, we denote the second fundamental form of <sup>M</sup> in <sup>M</sup><sup>~</sup> .

<sup>H</sup> <sup>¼</sup> <sup>1</sup> <sup>n</sup> traceðh<sup>Þ</sup> is called the mean curvature vector of <sup>M</sup> in <sup>M</sup><sup>~</sup> . If <sup>H</sup> <sup>¼</sup> 0, the sub-manifold is called minimal.

On the other hand, M is called pseudo-umbilical if there exists a function λ on M, such that

$$
\tilde{\mathcal{g}}\left(h(\mathbf{X}, \mathbf{Y}), H\right) = \lambda \mathbf{g}(\mathbf{X}, \mathbf{Y}) \tag{13}
$$

for any vector fields X, Y on M and M is called totally umbilical sub-manifold if

$$h(\mathbf{X}, \mathbf{Y}) = \mathcal{g}(\mathbf{X}, \mathbf{Y}) \mathbf{H}. \tag{14}$$

It is clear that every minimal sub-manifold is pseudo-umbilical with λ ¼ 0. On the other hand, by a direct calculation, we can find λ ¼ ~gðH, HÞ for a pseudo-umbilical sub-manifold. So, every totally umbilical sub-manifold is a pseudo-umbilical and a totally umbilical sub-manifold is totally geodesic if and only if it is minimal [2].

Now, let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g<sup>Þ</sup> and <sup>V</sup> be a normal vector field on M, X be a vector field on M. Then, we decompose

$$
\tilde{\nabla}\_X V = -A\_V X + \nabla\_X^\perp V,\tag{15}
$$

where AVX and ∇<sup>⊥</sup> <sup>X</sup>V denote the tangential and the normal components of ∇⊥ <sup>X</sup>V, respectively. We can easily see that AVX and ∇⊥ <sup>X</sup>V are both differentiable vector fields on M and normal bundle of M, respectively. Moreover, Eq. ð15Þ is also called Weingarten formula.

Proposition 1.2. Let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ. Then

(a) AVX is bilinear in vector fields V and X. Hence, AVX at point p∈M depends only on vector fields Vp and Xp.

(b) For any normal vector field V on M, we have

$$\lg(A\_V X, Y) = \lg\left(h(X, Y), V\right). \tag{16}$$

Proof: Let α and β be any two functions on M. Then, we have

$$
\begin{aligned}
\tilde{\nabla}\_{\alpha X}(\beta V) &= \alpha \tilde{\nabla}\_X(\beta V) \\&= \alpha \{ X(\beta)V + \beta \tilde{\nabla}\_X V \}
\end{aligned}
$$

$$
$$

This implies that

$$A\_{\beta} \alpha X = \alpha \beta A\_{V} X \tag{18}$$

and

$$
\nabla\_{\alpha\overline{\lambda}}^{\perp} \beta V = \alpha X(\beta) V + \alpha \beta \nabla\_X^{\perp} V. \tag{19}
$$

Thus, AVX is bilinear in V and X. Additivity is trivial. On the other hand, since gis a Riemannian metric,

$$X\_{\mathcal{S}}^{\sim}(Y, V) = 0,\tag{20}$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>Þ</sup> and <sup>V</sup>∈ΓðT<sup>⊥</sup>MÞ.

Eq. (12) implies that

$$
\tilde{\mathcal{g}}(\tilde{\nabla}\_X Y, V) + \tilde{\mathcal{g}}(Y, \tilde{\nabla}\_X V) = 0. \tag{21}
$$

By means of Eqs. (4) and (15), we obtain

Sub-Manifolds of a Riemannian Manifold http://dx.doi.org/10.5772/65948 51

$$
\tilde{\mathcal{g}}\left(h(X,Y),V\right) - \mathcal{g}(A\_V X, Y) = 0.\tag{22}
$$

The proof is completed [3].

Let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ, and <sup>h</sup> and AV denote the second fundamental form and shape operator of M, respectively.

The covariant derivative of h and AV is, respectively, defined by

$$h(\tilde{\nabla} \chi h)(Y, Z) = \nabla\_X^{\perp} h(Y, Z) - h(\nabla \chi Y, Z) - h(Y, \nabla \chi Z) \tag{23}$$

and

totally umbilical sub-manifold is a pseudo-umbilical and a totally umbilical sub-manifold is

Now, let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g<sup>Þ</sup> and <sup>V</sup> be a normal vector field

<sup>∇</sup><sup>~</sup> XV <sup>¼</sup> <sup>−</sup>AVX <sup>þ</sup> <sup>∇</sup><sup>⊥</sup>

bundle of M, respectively. Moreover, Eq. ð15Þ is also called Weingarten formula. Proposition 1.2. Let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>gÞ. Then

gðAVX,YÞ ¼ g

<sup>∇</sup><sup>~</sup> <sup>α</sup><sup>X</sup>ðβVÞ ¼ <sup>α</sup>∇<sup>~</sup> <sup>X</sup>ðβV<sup>Þ</sup>

<sup>X</sup>V denote the tangential and the normal components of ∇⊥

(a) AVX is bilinear in vector fields V and X. Hence, AVX at point p∈M depends only on vector

hðX, YÞ, V

<sup>¼</sup> <sup>α</sup>fXðβÞ<sup>V</sup> <sup>þ</sup> <sup>β</sup>∇<sup>~</sup> XV<sup>g</sup>

<sup>α</sup><sup>X</sup>β<sup>V</sup> <sup>¼</sup> <sup>α</sup>XðβÞV−αβAVX <sup>þ</sup> αβ∇<sup>⊥</sup>

<sup>α</sup><sup>X</sup>β<sup>V</sup> <sup>¼</sup> <sup>α</sup>XðβÞ<sup>V</sup> <sup>þ</sup> αβ∇⊥

Thus, AVX is bilinear in V and X. Additivity is trivial. On the other hand, since gis a Riemann-

<sup>X</sup>V, (15)

: (16)

<sup>X</sup>V: (17)

<sup>X</sup>V: (19)

<sup>X</sup>V are both differentiable vector fields on M and normal

Aβ<sup>V</sup>αX ¼ αβAVX (18)

X~gðY, VÞ ¼ 0, (20)

<sup>~</sup>gð∇<sup>~</sup> XY, <sup>V</sup>Þ þ <sup>~</sup>gðY, <sup>∇</sup><sup>~</sup> XVÞ ¼ <sup>0</sup>: (21)

<sup>X</sup>V, respectively.

totally geodesic if and only if it is minimal [2].

where AVX and ∇<sup>⊥</sup>

50 Manifolds - Current Research Areas

fields Vp and Xp.

This implies that

and

ian metric,

Eq. (12) implies that

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>Þ</sup> and <sup>V</sup>∈ΓðT<sup>⊥</sup>MÞ.

By means of Eqs. (4) and (15), we obtain

We can easily see that AVX and ∇⊥

on M, X be a vector field on M. Then, we decompose

(b) For any normal vector field V on M, we have

Proof: Let α and β be any two functions on M. Then, we have

<sup>−</sup>Aβ<sup>V</sup>α<sup>X</sup> <sup>þ</sup> ∇⊥

∇⊥

$$(\nabla\_X A)\_V Y = \nabla\_X (A\_V Y) - A\_{\nabla\_X^\perp V} Y - A\_V \nabla\_X Y \tag{24}$$

for any vector fields X, Y tangent to M and any vector field V normal to M. If ∇Xh ¼ 0 for all X, then the second fundamental form of M is said to be parallel, which is equivalent to ∇XA ¼ 0. By direct calculations, we get the relation

$$\mathcal{g}\left( (\nabla\_X h)(Y, Z), V \right) = \mathcal{g}\left( (\nabla\_X A)\_V Y, Z \right). \tag{25}$$

Example 1.1. We consider the isometric immersion

$$
\phi \!\!\!/ \!\!/ \!\!\!/ \!\!\!/ \!\!/ \!\!\!/ \!\!\/? \!\!\/ \!\!\/? \!\!\/? \tag{26}
$$

$$\phi(\mathbf{x}\_1, \mathbf{x}\_2) = (\mathbf{x}\_1, \sqrt{\mathbf{x}\_1^2 - 1}, \mathbf{x}\_2, \sqrt{\mathbf{x}\_2^2 - 1}) \tag{27}$$

we note that <sup>M</sup> <sup>¼</sup> <sup>φ</sup>ðR<sup>2</sup>Þ⊂R<sup>4</sup> is a two-dimensional sub-manifold of <sup>R</sup><sup>4</sup> and the tangent bundle is spanned by the vectors

$$TM = \mathbb{S}\_{\mathbb{P}}\left\{\mathbf{e}\_1 = \left(\sqrt{\mathbf{x}\_1^2 - 1}, \mathbf{x}\_1, 0, 0\right), \mathbf{e}\_2 = \left(0, 0, \sqrt{\mathbf{x}\_2^2 - 1}, \mathbf{x}\_2\right)\right\} \text{ and the normal vector fields}$$

$$T^{\perp}M = sp\left\{\mathbf{w}\_1 = \left(-\mathbf{x}\_1, \sqrt{\mathbf{x}\_1^2 - 1}, 0, 0\right), \mathbf{w}\_2 = \left(0, 0, -\mathbf{x}\_1, \sqrt{\mathbf{x}\_2^2 - 1}\right)\right\}. \tag{28}$$

By ∇~, we denote the Levi-Civita connection of R<sup>4</sup> , the coefficients of connection, are given by

$$
\check{\nabla}\_{e\_1} e\_1 = \frac{2\mathbf{x}\_1 \sqrt{\mathbf{x}\_1^2 - 1}}{2\mathbf{x}\_1^2 - 1} e\_1 - \frac{1}{2\mathbf{x}\_1^2 - 1} w\_1,\tag{29}
$$

$$
\tilde{\nabla}\_{c\_2} e\_2 = \frac{2\chi\_2\sqrt{\chi\_2^2 - 1}}{2\chi\_2^2 - 1} e\_2 - \frac{1}{2\chi\_2^2 - 1} w\_2 \tag{30}
$$

and

$$
\nabla\_{e\_2} e\_1 = 0.\tag{31}
$$

Thus, we have <sup>h</sup>ðe1,e1Þ ¼ <sup>−</sup> <sup>1</sup> 2x<sup>2</sup> <sup>1</sup>−<sup>1</sup> <sup>w</sup>1, <sup>h</sup>ðe2,e2Þ ¼ <sup>−</sup> <sup>1</sup> 2x<sup>2</sup> <sup>2</sup>−<sup>1</sup> <sup>w</sup><sup>2</sup> and <sup>h</sup>ðe2,e1Þ ¼ <sup>0</sup>: The mean curvature vector of <sup>M</sup> <sup>¼</sup> <sup>φ</sup>ðR<sup>2</sup> Þ is given by

$$H = -\frac{1}{2}(w\_1 + w\_2). \tag{32}$$

Furthermore, by using Eq. (16), we obtain

$$\begin{aligned} \operatorname{g}(A\_{w\_1}e\_1, e\_1) &= \operatorname{g}\left(h(e\_1, e\_1), \varpi\_1\right) = -\frac{1}{2\chi\_1^2 - 1} (\chi\_1^2 + \chi\_1^2 - 1) = -1, \\ \operatorname{g}(A\_{w\_1}e\_2, e\_2) &= \operatorname{g}\left(h(e\_2, e\_2), \varpi\_1\right) = -\frac{1}{2\chi\_2^2 - 1} \operatorname{g}(w\_1, w\_2) = 0, \\ \operatorname{g}(A\_{w\_1}e\_1, e\_2) &= 0, \end{aligned} \tag{33}$$

and

$$\begin{aligned} \mathcal{g}(A\_{w\_2}e\_1, e\_1) &= \mathcal{g}\left(h(e\_1, e\_1), w\_2\right) = 0, \\ \mathcal{g}(A\_{w\_2}e\_1, e\_2) &= 0, \mathcal{g}(A\_{w\_2}e\_2, e\_2) = 1. \end{aligned} \tag{34}$$

Thus, we have

$$A\_{w\_1} = \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix} \text{ and } A\_{w\_2} = \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix}. \tag{35}$$

Now, let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>g</sup>Þ, <sup>R</sup><sup>~</sup> and <sup>R</sup> be the Riemannian curvature tensors of M~ and M, respectively. From then the Gauss and Weingarten formulas, we have

$$\begin{split} \tilde{R}(X,Y)Z &= \tilde{\nabla}\_X \tilde{\nabla}\_Y Z - \tilde{\nabla}\_Y \tilde{\nabla}\_X Z - \tilde{\nabla}\_{[X,Y]} Z \\ &= \tilde{\nabla}\_X \Big( \nabla\_Y Z + h(Y,Z) \Big) - \tilde{\nabla}\_Y \Big( \nabla\_X Z + h(X,Z) \Big) - \nabla\_{[X,Y]} Z - h([X,Y],Z) \\ &= \tilde{\nabla}\_X \nabla\_Y Z + \tilde{\nabla}\_X h(Y,Z) - \tilde{\nabla}\_Y \nabla\_X Z - \tilde{\nabla}\_Y h(X,Z) - \nabla\_{[X,Y]} Z - h(\nabla\_X Y, Z) + h(\nabla\_Y X, Z) \\ &= \nabla\_X \nabla\_Y Z - \nabla\_Y \nabla\_X Z + h(X, \nabla\_Y Z) - h(\nabla\_X Z, Z) + \nabla\_X^2 h(Y,Z) \\ &\quad - A\_{h(Y,Z)} X - \nabla\_Y^2 h(X,Z) + A\_{h(X,Z)} Y - \nabla\_{[X,Y]} Z - h(\nabla\_X Y, Z) + h(\nabla\_Y X, Z) \\ &= \nabla\_X \nabla\_Y Z - \nabla\_Y \nabla\_X Z - \nabla\_{[X,Y]} Z - \nabla\_Y^2 h(Y,Z) - h(\nabla\_X Y, Z) \\ &\quad - h(Y, \nabla\_X Z) - \nabla\_Y^2 h(X, Z) + h(\nabla\_Y X, Z) + h(\nabla\_Y Z, X) \\ &\quad + A\_{h(X,Z)} Y - A\_{h(Y,Z)} X \\ &= R(X, Y)Z + (\nabla\_X h)(Y, Z) - (\nabla\_Y h)(X, Z) + A\_{h(X,Z)} Y - A\_{h(Y,Z)} X \end{split}$$

from which

∇<sup>e</sup><sup>2</sup> e<sup>1</sup> ¼ 0: (31)

<sup>2</sup>−<sup>1</sup> <sup>w</sup><sup>2</sup> and <sup>h</sup>ðe2,e1Þ ¼ <sup>0</sup>: The mean curvature

ðw<sup>1</sup> þ w2Þ: (32)

<sup>1</sup>−1Þ ¼ −1,

(33)

gðw1, w2Þ ¼ 0,

and

and

have

Thus, we have

Thus, we have <sup>h</sup>ðe1,e1Þ ¼ <sup>−</sup> <sup>1</sup>

vector of <sup>M</sup> <sup>¼</sup> <sup>φ</sup>ðR<sup>2</sup>

52 Manifolds - Current Research Areas

2x<sup>2</sup>

Þ is given by

gðAw<sup>1</sup> e1,e1Þ ¼ g

gðAw<sup>1</sup> e2,e2Þ ¼ g

gðAw<sup>1</sup> e1,e2Þ ¼ 0,

Aw<sup>1</sup> ¼

<sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> <sup>¼</sup> <sup>∇</sup><sup>~</sup> <sup>X</sup>∇<sup>~</sup> YZ−∇<sup>~</sup> <sup>Y</sup>∇<sup>~</sup> XZ−∇<sup>~</sup> <sup>½</sup>X,Y�<sup>Z</sup>

∇YZ þ hðY, ZÞ

<sup>−</sup>AhðY,ZÞX−∇<sup>⊥</sup>

<sup>−</sup>hðY, <sup>∇</sup>XZÞ−∇⊥

þAhðX,ZÞY−AhðY,ZÞX

<sup>¼</sup> <sup>∇</sup><sup>~</sup> <sup>X</sup> 

Furthermore, by using Eq. (16), we obtain

<sup>1</sup>−<sup>1</sup> <sup>w</sup>1, <sup>h</sup>ðe2,e2Þ ¼ <sup>−</sup> <sup>1</sup>

H ¼ − 1 2

hðe1,e1Þ, w<sup>1</sup>

hðe2,e2Þ, w<sup>1</sup>

gðAw<sup>2</sup> e1,e1Þ ¼ g

 −1 0 0 0

> −∇~ <sup>Y</sup>

<sup>¼</sup> <sup>∇</sup>X∇YZ−∇Y∇XZ−∇½X,Y�<sup>Z</sup> <sup>þ</sup> <sup>∇</sup><sup>⊥</sup>

<sup>¼</sup> <sup>∇</sup>X∇YZ−∇Y∇XZ <sup>þ</sup> <sup>h</sup>ðX, <sup>∇</sup>YZÞ−hð∇XZ, <sup>Y</sup>Þ þ <sup>∇</sup><sup>⊥</sup>

2x<sup>2</sup>

<sup>¼</sup> <sup>−</sup> <sup>1</sup> 2x<sup>2</sup> <sup>1</sup>−1 ðx2 <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup>

<sup>¼</sup> <sup>−</sup> <sup>1</sup> 2x<sup>2</sup> <sup>2</sup>−1

hðe1,e1Þ, w<sup>2</sup>

and Aw<sup>2</sup> ¼

∇XZ þ hðX, ZÞ

<sup>¼</sup> <sup>∇</sup><sup>~</sup> <sup>X</sup>∇YZ <sup>þ</sup> <sup>∇</sup><sup>~</sup> XhðY, <sup>Z</sup>Þ−∇<sup>~</sup> <sup>Y</sup>∇XZ−∇<sup>~</sup> YhðX, <sup>Z</sup>Þ−∇½X,Y�Z−hð∇XY, <sup>Z</sup>Þ þ <sup>h</sup>ð∇YX, <sup>Z</sup><sup>Þ</sup>

<sup>Y</sup>hðX, ZÞ þ hð∇YX,ZÞ þ hð∇YZ, XÞ

¼ RðX, YÞZ þ ð∇XhÞðY, ZÞ−ð∇YhÞðX, ZÞ þ AhðX,ZÞY−AhðY,ZÞX (36)

Now, let <sup>M</sup> be a sub-manifold of a Riemannian manifold <sup>ð</sup>M<sup>~</sup> , <sup>g</sup>Þ, <sup>R</sup><sup>~</sup> and <sup>R</sup> be the Riemannian curvature tensors of M~ and M, respectively. From then the Gauss and Weingarten formulas, we

 ¼ 0, <sup>g</sup>ðAw<sup>2</sup> <sup>e</sup>1,e2Þ ¼ <sup>0</sup>, <sup>g</sup>ðAw<sup>2</sup> <sup>e</sup>2,e2Þ ¼ <sup>1</sup>: (34)

> 0 0 0 −1

YhðX, ZÞ þ AhðX,ZÞY−∇½X,Y�Z−hð∇XY, ZÞ þ hð∇YX, ZÞ

XhðY, ZÞ−hð∇XY, ZÞ

−∇½X,Y�Z−hð½X, Y�, ZÞ

<sup>X</sup>hðY, ZÞ

: (35)

$$\bar{R}(X,Y)Z = R(X,Y)Z + A\_{h(X,Z)}Y - A\_{h(Y,Z)}X + (\nabla \chi h)(Y,Z) - (\nabla\_Y h)(X,Z),\tag{37}$$

for any vector fields X, Y and Z tangent to M. For any vector field W tangent to M, Eq. (37) gives the Gauss equation

$$\operatorname{g}\left(\tilde{\mathcal{R}}(X,Y)Z,W\right) = \operatorname{g}\left(\mathcal{R}(X,Y)Z,W\right) + \operatorname{g}\left(h(Y,W),h(X,Z)\right) - \operatorname{g}\left(h(Y,Z),h(X,W)\right). \tag{38}$$

On the other hand, the normal component of Eq. (37) is called equation of Codazzi, which is given by

$$\left(\tilde{R}(X,Y)Z\right)^{\perp} = (\nabla\_X h)(Y,Z) - (\nabla\_Y h)(X,Z). \tag{39}$$

If the Codazzi equation vanishes identically, then sub-manifold M is said to be curvatureinvariant sub-manifold [4].

In particular, if <sup>M</sup><sup>~</sup> is of constant curvature, <sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> is tangent to <sup>M</sup>, that is, sub-manifold is curvature-invariant. Whereas, in Kenmotsu space forms, and Sasakian space forms, this not true.

Next, we will define the curvature tensor R<sup>⊥</sup> of the normal bundle of the sub-manifold M by

$$R^\perp(X,Y)V = \nabla\_X^\perp \nabla\_Y^\perp V - \nabla\_Y^\perp \nabla\_X^\perp V - \nabla\_{[X,Y]}^\perp V \tag{40}$$

for any vector fields X, Y tangent to sub-manifold M, and any vector field V normal to M. From the Gauss and Weingarten formulas, we have

$$\begin{split} \hat{R}(X,Y)V &= \hat{\nabla}\_X \hat{\nabla}\_Y V - \hat{\nabla}\_Y \hat{\nabla}\_X V - \hat{\nabla}\_{[X,Y]} V \\ &= \hat{\nabla}\_X (-A\_V Y + \nabla\_Y^\perp V) - \hat{\nabla}\_Y (-A\_V X + \nabla\_X^\perp V) + A\_V[X,Y] - \nabla\_{[X,Y]}^\perp V \\ &= -\hat{\nabla}\_X A\_V Y + \hat{\nabla}\_Y A\_V X + \hat{\nabla}\_X \nabla\_Y^\perp V - \hat{\nabla}\_Y \nabla\_X^\perp V + A\_V[X,Y] - \nabla\_{[X,Y]}^\perp V \\ &= -\nabla\_X A\_V Y - h(X, A\_V Y) + \nabla\_Y A\_V X + h(Y, A\_V X) \\ &+ \nabla\_X^\perp \nabla\_Y^\perp V - \nabla\_Y^\perp \nabla\_X^\perp V - A\_{\nabla\_X^\perp V} X + A\_{\nabla\_X^\perp V} Y + A\_V[X,Y] - \nabla\_{[X,Y]}^\perp V \\ &= \nabla\_X^\perp \nabla\_Y V - \nabla\_Y^\perp \nabla\_X^\perp V - \nabla\_{[X,Y]}^\perp V - A\_{\nabla\_X^\perp V} X + A\_{\nabla\_X^\perp V} Y + A\_V[X,Y] \\ &- \nabla\_X A\_V Y + \nabla\_Y A\_V X - h(X, A\_V Y) + h(Y, A\_V X) \\ &= R^\perp(X,Y)V + h(A\_V X, Y) - h(X, A\_V Y) - (\nabla\_X A\_V)\_V Y + (\nabla\_Y A\_V X - h(X, A\_V Y)). \tag{41} \end{split}$$

For any normal vector U to M, we obtain

$$\begin{aligned} \operatorname{g}\left(\check{\mathbb{R}}\left(\mathbf{X},\mathbf{Y}\right)V,\mathcal{U}\right) &= \operatorname{g}\left(\mathbb{R}^{\perp}(\mathbf{X},\mathbf{Y})V,\mathcal{U}\right) + \operatorname{g}\left(h(A\_V\mathbf{X},\mathbf{Y}),\mathcal{U}\right) - \operatorname{g}\left(h(\mathbf{X},A\_V\mathbf{Y}),\mathcal{U}\right) \\ &= \operatorname{g}\left(\mathbb{R}^{\perp}(\mathbf{X},\mathbf{Y})V,\mathcal{U}\right) + \operatorname{g}\left(A\_{\mathcal{U}}\mathbf{Y},A\_V\mathbf{X}\right) - \operatorname{g}\left(A\_V\mathbf{Y},A\_{\mathcal{U}}\mathbf{X}\right) \\ &= \operatorname{g}\left(\mathbb{R}^{\perp}(\mathbf{X},\mathbf{Y})V,\mathcal{U}\right) + \operatorname{g}\left(A\_{\mathcal{V}}A\_{\mathcal{U}}\mathbf{Y},\mathcal{X}\right) - \operatorname{g}\left(A\_{\mathcal{U}}A\_{V}Y,\mathcal{X}\right) \end{aligned} \tag{42}$$

Since ½AU, AV� ¼ AUAV−AVAU, Eq. (42) implies

$$\mathcal{S}\left(\tilde{R}(X,Y)V,\mathcal{U}\right) = \mathcal{S}\left(R^{\perp}(X,Y)V,\mathcal{U}\right) + \mathcal{G}([A\_{\mathcal{U}},A\_{V}]Y,X).\tag{43}$$

Eq. (43) is also called the Ricci equation.

If <sup>R</sup><sup>⊥</sup> <sup>¼</sup> 0, then the normal connection of <sup>M</sup> is said to be flat [2].

When <sup>R</sup><sup>~</sup> <sup>ð</sup>X,YÞ<sup>V</sup> ⊥ ¼ 0, the normal connection of the sub-manifold M is flat if and only if the second fundamental form M is commutative, i.e. ½AU, AV� ¼ 0 for all U, V. If the ambient space <sup>M</sup><sup>~</sup> is real space form, then <sup>R</sup><sup>~</sup> <sup>ð</sup>X,YÞ<sup>V</sup> ⊥ ¼ 0 and hence the normal connection of M is flat if and only if the second fundamental form is commutative. If <sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> tangent to <sup>M</sup>, then equation of codazzi Eq. (37) reduces to

$$(\nabla \chi h)(Y, Z) = (\nabla\_Y h)(X, Z) \tag{44}$$

which is equivalent to

$$(\nabla\_X A)\_V Y = (\nabla\_Y A)\_V X. \tag{45}$$

On the other hand, if the ambient space M~ is a space of constant curvature c, then we have

$$
\tilde{R}(X,Y)Z = c\{\mathcal{g}(Y,Z)X \neg \mathcal{g}(X,Z)Y\} \tag{46}
$$

for any vector fields X, Y and Z on M~ .

Since <sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> is tangent to <sup>M</sup>, the equation of Gauss and the equation of Ricci reduce to

$$\begin{aligned} \text{g}\left(\text{R}(X,Y)Z,\mathcal{W}\right) &= \text{c}\{\text{g}(Y,Z)\text{g}(X,\mathcal{W})\text{-g}(X,Z)\text{g}(Y,\mathcal{W})\} \\ &+ \text{g}\left(\text{h}(Y,Z),\text{h}(X,\mathcal{W})\right) \text{-g}\left(\text{h}(Y,W),\text{h}(X,Z)\right) \end{aligned} \tag{47}$$

and

$$\mathcal{S}\left(\boldsymbol{R}^{\perp}(\boldsymbol{X},\boldsymbol{Y})\boldsymbol{V},\boldsymbol{U}\right) = \mathcal{S}([A\_{\boldsymbol{U}}, A\_{\boldsymbol{V}}]\boldsymbol{X},\boldsymbol{Y}),\tag{48}$$

respectively.

Proposition 1.3. A totally umbilical sub-manifold M in a real space form M~ of constant curvature c is also of constant curvature.

Proof: Since M is a totally umbilical sub-manifold of M~ of constant curvature c, by using Eqs. (14) and (46), we have

$$\mathcal{S}\left(\mathbb{R}(X,Y)Z,\mathcal{W}\right) = \mathfrak{c}\{\mathfrak{g}(Y,Z)\mathfrak{g}(X,\mathcal{W})\mathfrak{-g}(X,Z)\mathfrak{g}(Y,\mathcal{W})\}$$

$$\begin{split} & + \mathfrak{g}(H,H)\{\mathfrak{g}(Y,Z)\mathfrak{g}(X,\mathcal{W})\mathfrak{-g}(X,Z)\mathfrak{g}(Y,\mathcal{W})\} \\ & = \{\mathfrak{c} + \mathfrak{g}(H,H)\}\{\mathfrak{g}(Y,Z)\mathfrak{g}(X,\mathcal{W})\mathfrak{-g}(X,Z)\mathfrak{g}(Y,\mathcal{W})\}. \end{split} \tag{49}$$

This shows that the sub-manifold <sup>M</sup> is of constant curvature <sup>c</sup> <sup>þ</sup> ‖H<sup>2</sup> ‖ for n > 2. If n ¼ 2, ‖H‖ ¼ constant follows from the equation of Codazzi [3].

This proves the proposition.

g 

54 Manifolds - Current Research Areas

When 

and

respectively.

<sup>R</sup><sup>~</sup> <sup>ð</sup>X,YÞ<sup>V</sup>

M~ is real space form, then

which is equivalent to

of codazzi Eq. (37) reduces to

<sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>ÞV, <sup>U</sup>

 ¼ g 

Since ½AU, AV� ¼ AUAV−AVAU, Eq. (42) implies

g 

Eq. (43) is also called the Ricci equation.

⊥

for any vector fields X, Y and Z on M~ .

g 

RðX, YÞZ, W

g  þg 

<sup>R</sup><sup>⊥</sup>ðX, <sup>Y</sup>ÞV, <sup>U</sup>

¼ g 

¼ g 

<sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>ÞV, <sup>U</sup>

If <sup>R</sup><sup>⊥</sup> <sup>¼</sup> 0, then the normal connection of <sup>M</sup> is said to be flat [2].

<sup>R</sup><sup>~</sup> <sup>ð</sup>X,YÞ<sup>V</sup>

⊥

<sup>R</sup><sup>⊥</sup>ðX, <sup>Y</sup>ÞV, <sup>U</sup>

<sup>R</sup><sup>⊥</sup>ðX, <sup>Y</sup>ÞV, <sup>U</sup>

<sup>R</sup><sup>⊥</sup>ðX, <sup>Y</sup>ÞV, <sup>U</sup>

 ¼ g   þ g 

<sup>R</sup><sup>⊥</sup>ðX, <sup>Y</sup>ÞV, <sup>U</sup>

second fundamental form M is commutative, i.e. ½AU, AV� ¼ 0 for all U, V. If the ambient space

only if the second fundamental form is commutative. If <sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> tangent to <sup>M</sup>, then equation

On the other hand, if the ambient space M~ is a space of constant curvature c, then we have

Since <sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> is tangent to <sup>M</sup>, the equation of Gauss and the equation of Ricci reduce to

¼ 0, the normal connection of the sub-manifold M is flat if and only if the

¼ 0 and hence the normal connection of M is flat if and

ð∇XhÞðY, ZÞ¼ð∇YhÞðX, ZÞ (44)

ð∇XAÞVY ¼ ð∇YAÞVX: (45)

<sup>R</sup><sup>~</sup> <sup>ð</sup>X, <sup>Y</sup>Þ<sup>Z</sup> <sup>¼</sup> <sup>c</sup>fgðY, <sup>Z</sup>ÞX−gðX, <sup>Z</sup>ÞY<sup>g</sup> (46)

¼ cfgðY, ZÞgðX, WÞ−gðX,ZÞgðY, WÞg

 −g 

hðY, WÞ, hðX, ZÞ

¼ gð½AU, AV�X, YÞ, (48)

(47)

hðY, ZÞ, hðX, WÞ

hðAVX, YÞ, U

 −g 

þ gðAVAUY, XÞ−gðAUAVY, XÞ (42)

þ gðAUY, AVXÞ−gðAVY, AUXÞ

hðX, AVYÞ, U

þ gð½AU, AV�Y, XÞ: (43)

On the other hand, for any orthonormal basis feag of normal space, we have

$$\begin{aligned} \operatorname{g}(Y,Z)\operatorname{g}(X,W)\operatorname{-g}(X,Z)\operatorname{g}(Y,W) &= \sum\_{a} \Big[ \operatorname{g}\Big(h(Y,Z),e\_{a}\Big) \operatorname{g}\Big(h(X,W),e\_{a}\Big) \\ &- \operatorname{g}\Big(h(X,Z),e\_{a}\Big) \operatorname{g}\Big(h(Y,W),e\_{a}\Big) \\ &= \sum\_{a} \operatorname{g}(A\_{e\_{a}}Y,Z)\operatorname{g}(A\_{e\_{a}}X,W)\operatorname{-g}(A\_{e\_{a}}X,Z)\operatorname{g}(A\_{e\_{a}}Y,W) \end{aligned} \tag{50}$$

Thus, Eq. (45) can be rewritten as

$$\begin{split} \log \left( R(X,Y)Z,\mathcal{W} \right) &= c \{ \mathbf{g}(Y,Z)\mathbf{g}(X,\mathcal{W}) \neg \mathbf{g}(X,Z)\mathbf{g}(Y,\mathcal{W}) \} \\ &+ \sum\_{a} [\mathbf{g}(A\_{e\_{a}}Y,Z)\mathbf{g}(A\_{e\_{a}}X,\mathcal{W}) \neg \mathbf{g}(A\_{e\_{a}}X,Z)\mathbf{g}(A\_{e\_{a}}Y,\mathcal{W})] \end{split} \tag{51}$$

By using Aea , we can construct a similar equation to Eq. (47) for Eq. (23).

Now, let S- be the Ricci tensor of M. Then, Eq. (47) gives us

$$S(X,Y) = \mathfrak{c}\{\mathfrak{ng}(X,Y) \text{--} \mathfrak{g}(e\_i, X)\mathfrak{g}(e\_i, Y)\}\tag{52}$$

$$+\sum\_{e\_{\varepsilon}} [\mathbb{g}(A\_{e\_{\varepsilon}}e\_i, e\_i)\mathbb{g}(A\_{e\_{\varepsilon}}X, Y)\cdots\mathbb{g}(A\_{e\_{\varepsilon}}X, e\_i)\mathbb{g}(A\_{e\_{\varepsilon}}e\_i, Y)]$$

$$=c(n-1)\mathbb{g}(X, Y) + \sum\_{e\_{\varepsilon}} [Tr(A\_{e\_{\varepsilon}})\mathbb{g}(A\_{e\_{\varepsilon}}X, Y)\cdots\mathbb{g}(A\_{e\_{\varepsilon}}X, A\_{e\_{\varepsilon}}Y)],\tag{53}$$

where fe1,e2, …,eng are orthonormal basis of M.

Therefore, the scalar curvature r of sub-manifold M is given by

$$r = c n(n-1) \sum\_{e\_d} Tr^2(A\_{e\_d}) - \sum\_{e\_d} Tr(A\_{e\_d})^2 \tag{54}$$

∑ ea TrðAea Þ <sup>2</sup> is the square of the length of the second fundamental form of M, which is denoted by jAea j 2 . Thus, we also have

$$\|h^2\| = \sum\_{i,j=1}^n \text{g}\left(h(e\_i, e\_j), h(e\_i, e\_j)\right) = \|A^2\|.\tag{55}$$
