2. Distribution on a manifold

An m-dimensional distribution on a manifold M~ is a mapping D defined on M~ , which assignes to each point <sup>p</sup> of <sup>M</sup><sup>~</sup> an <sup>m</sup>-dimensional linear subspace <sup>D</sup><sup>p</sup> of TM<sup>~</sup> <sup>ð</sup>pÞ. A vector field <sup>X</sup> on <sup>M</sup><sup>~</sup> belongs to <sup>D</sup> if we have Xp∈D<sup>p</sup> for each <sup>p</sup>∈M<sup>~</sup> . When this happens, we write <sup>X</sup>∈ΓðDÞ. The distribution D is said to be differentiable if for any p∈M~ , there exist m-differentiable linearly independent vector fields Xj∈ΓðDÞ in a neighbordhood of p.

The distribution D is said to be involutive if for all vector fields X, Y∈ΓðDÞ we have <sup>½</sup>X,Y�∈ΓðDÞ. A sub-manifold <sup>M</sup> of <sup>M</sup><sup>~</sup> is said to be an integral manifold of <sup>D</sup> if for every point p∈M, D<sup>p</sup> coincides with the tangent space to M at p. If there exists no integral manifold of D which contains M, then M is called a maximal integral manifold or a leaf of D. The distribution D is said to be integrable if for every p∈M~ , there exists an integral manifold of D containing p [2].

Let ∇~ and distribution be a linear connection on M~ , respectively. The distribution D is said to be parallel with respect to M~ , if we have

$$\tilde{\nabla}\_X \varGamma \in \Gamma(\mathcal{D}) \text{ for all } X \in \Gamma(T\tilde{M}) \text{ and } \varGamma \in \Gamma(\mathcal{D}) \tag{56}$$

Now, let <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g<sup>Þ</sup> be Riemannian manifold and <sup>D</sup> be a distribution on <sup>M</sup><sup>~</sup> . We suppose <sup>M</sup><sup>~</sup> is endowed with two complementary distribution <sup>D</sup> and <sup>D</sup><sup>⊥</sup>, i:e:, we have TM<sup>~</sup> <sup>¼</sup> <sup>D</sup>⊕D<sup>⊥</sup>. Denoted by P and Q the projections of TM~ to D and D<sup>⊥</sup>, respectively.

Theorem 2.1. All the linear connections with respect to which both distributions D and D<sup>⊥</sup> are parallel, are given by

$$
\nabla\_X Y = P\nabla\_X' PY + Q\nabla\_X' QY + PS(X, PY) + QS(X, QY) \tag{57}
$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>, where <sup>∇</sup><sup>0</sup> and S are, respectively, an arbitrary linear connection and arbitrary tensor field of type <sup>ð</sup>1, <sup>2</sup><sup>Þ</sup> on <sup>M</sup><sup>~</sup> .

Proof: Suppose ∇<sup>0</sup> is an arbitrary linear connection on M~ . Then, any linear connection ∇ on M~ is given by

$$
\nabla\_X Y = \nabla\_X^{'} Y + S(X, Y) \tag{58}
$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>. We can put

<sup>r</sup> <sup>¼</sup> cnðn−1Þ∑

‖h<sup>2</sup>

independent vector fields Xj∈ΓðDÞ in a neighbordhood of p.

‖ <sup>¼</sup> ∑ n

<sup>i</sup>, <sup>j</sup>¼<sup>1</sup> g 

∑ ea

TrðAea Þ

56 Manifolds - Current Research Areas

containing p [2].

parallel, are given by

Proof: Suppose ∇<sup>0</sup>

is given by

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>, where <sup>∇</sup><sup>0</sup>

arbitrary tensor field of type <sup>ð</sup>1, <sup>2</sup><sup>Þ</sup> on <sup>M</sup><sup>~</sup> .

. Thus, we also have

2. Distribution on a manifold

be parallel with respect to M~ , if we have

by jAea j 2 ea Tr<sup>2</sup> <sup>ð</sup>Aea <sup>Þ</sup>−∑ ea

<sup>2</sup> is the square of the length of the second fundamental form of M, which is denoted

hðei,ejÞ, hðei,ejÞ

An m-dimensional distribution on a manifold M~ is a mapping D defined on M~ , which assignes to each point <sup>p</sup> of <sup>M</sup><sup>~</sup> an <sup>m</sup>-dimensional linear subspace <sup>D</sup><sup>p</sup> of TM<sup>~</sup> <sup>ð</sup>pÞ. A vector field <sup>X</sup> on <sup>M</sup><sup>~</sup> belongs to <sup>D</sup> if we have Xp∈D<sup>p</sup> for each <sup>p</sup>∈M<sup>~</sup> . When this happens, we write <sup>X</sup>∈ΓðDÞ. The distribution D is said to be differentiable if for any p∈M~ , there exist m-differentiable linearly

The distribution D is said to be involutive if for all vector fields X, Y∈ΓðDÞ we have <sup>½</sup>X,Y�∈ΓðDÞ. A sub-manifold <sup>M</sup> of <sup>M</sup><sup>~</sup> is said to be an integral manifold of <sup>D</sup> if for every point p∈M, D<sup>p</sup> coincides with the tangent space to M at p. If there exists no integral manifold of D which contains M, then M is called a maximal integral manifold or a leaf of D. The distribution D is said to be integrable if for every p∈M~ , there exists an integral manifold of D

Let ∇~ and distribution be a linear connection on M~ , respectively. The distribution D is said to

Now, let <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g<sup>Þ</sup> be Riemannian manifold and <sup>D</sup> be a distribution on <sup>M</sup><sup>~</sup> . We suppose <sup>M</sup><sup>~</sup> is endowed with two complementary distribution <sup>D</sup> and <sup>D</sup><sup>⊥</sup>, i:e:, we have TM<sup>~</sup> <sup>¼</sup> <sup>D</sup>⊕D<sup>⊥</sup>.

Theorem 2.1. All the linear connections with respect to which both distributions D and D<sup>⊥</sup> are

Denoted by P and Q the projections of TM~ to D and D<sup>⊥</sup>, respectively.

<sup>X</sup>PY <sup>þ</sup> <sup>Q</sup>∇<sup>0</sup>

<sup>∇</sup>XY <sup>¼</sup> <sup>P</sup>∇<sup>0</sup>

<sup>∇</sup><sup>~</sup> XY∈ΓðD<sup>Þ</sup> for all <sup>X</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup> and <sup>Y</sup>∈ΓðD<sup>Þ</sup> (56)

<sup>X</sup>QY þ PSðX, PYÞ þ QSðX, QYÞ (57)

and S are, respectively, an arbitrary linear connection and

is an arbitrary linear connection on M~ . Then, any linear connection ∇ on M~

TrðAea Þ

 <sup>¼</sup> ‖A<sup>2</sup>

<sup>2</sup> (54)

‖: (55)

$$X = PX + QX \tag{59}$$

for any <sup>X</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>. Then, we have

$$
\nabla\_X Y = \nabla\_X (PY + QY) = \nabla\_X PY + \nabla\_X QY = \nabla\_X PY + S(X, PY)
$$

$$
+ \nabla\_X' QY + S(X, QY) = \mathbf{P}\nabla\_X' PY + Q\nabla\_X' PY + PS(X, PY) + QS(X, PY)
$$

$$
+ P\nabla\_X' QY + Q\nabla\_X' QY + PS(X, QY) + QS(X, QY) \tag{60}
$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>.

The distributions D and D are both parallel with respect to ∇ if and only if we have

$$
\phi(\nabla\_X PY) = 0 \\
\text{and} \\
\mathbf{P}(\nabla\_X QY) = 0. \tag{61}
$$

From Eqs. (58) and (61), it follows that D and D<sup>⊥</sup> are parallel with respect to ∇ if and only if

$$\text{Q}\overline{\text{V}}\_{X}^{\prime}PY + \text{Q}\text{S}(X, PY) = 0 \text{ and } \text{P}\overline{\text{V}}\_{X}^{\prime}QY + \text{PS}(X, QY) = 0. \tag{62}$$

Thus, Eqs. (58) and (62) give us Eq. (57).

Next, by means of the projections <sup>P</sup> and <sup>Q</sup>, we define a tensor field <sup>F</sup> of type <sup>ð</sup>1, <sup>1</sup><sup>Þ</sup> on <sup>M</sup><sup>~</sup> by

$$FX = PX \text{--} QX \tag{63}$$

for any <sup>X</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>. By a direct calculation, it follows that <sup>F</sup><sup>2</sup> <sup>¼</sup> <sup>I</sup>. Thus, we say that <sup>F</sup> defines an almost product structure on M~ . The covariant derivative of F is defined by

$$(\nabla\_X F)Y = \nabla\_X FY - \mathcal{F}\nabla\_X Y \tag{64}$$

for all <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>. We say that the almost product structure <sup>F</sup> is parallel with respect to the connection ∇, if we have ∇XF ¼ 0. In this case, F is called the Riemannian product structure [2].

Theorem 2.2. Let <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g<sup>Þ</sup> be a Riemannian manifold and <sup>D</sup>, <sup>D</sup><sup>⊥</sup> be orthogonal distributions on <sup>M</sup><sup>~</sup> such that TM<sup>~</sup> <sup>¼</sup> <sup>D</sup>⊕D<sup>⊥</sup>: Both distributions <sup>D</sup> and <sup>D</sup><sup>⊥</sup> are parallel with respect to <sup>∇</sup> if and only if F is a Riemannian product structure.

Proof: For any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup>, we can write

$$
\tilde{\nabla}\_Y PX = \tilde{\nabla}\_{PY} PX + \tilde{\nabla}\_{\tilde{Q}Y} PX \tag{65}
$$

and

$$
\tilde{\nabla}\_Y X = \tilde{\nabla}\_{PY} PX + \tilde{\nabla}\_{PY} QX + \tilde{\nabla}\_{QY} PX + \tilde{\nabla}\_{QY} QX,\tag{66}
$$

from which

$$\operatorname{g}(\stackrel{\sim}{\nabla}\_{\mathcal{Q}Y} \mathcal{P}X, \mathcal{Q}Z) = \mathcal{Q}Y\mathcal{g}(\mathcal{P}X, \mathcal{Q}Z) - \mathcal{g}(\nabla\_{\mathcal{Q}Y} \mathcal{Q}Z, \mathcal{P}X) = 0 - \mathcal{g}(\stackrel{\sim}{\nabla}\_{\mathcal{Q}Y} \mathcal{Q}Z, \mathcal{P}X) = 0,\tag{67}$$

that is, <sup>∇</sup>QYPX∈ΓðD<sup>Þ</sup> and so <sup>P</sup>∇<sup>~</sup> QYPX <sup>¼</sup> <sup>∇</sup><sup>~</sup> QYPX,

$$\mathbb{Q}\tilde{\nabla}\_{\mathcal{Q}Y}PX = 0.\tag{68}$$

In the same way, we obtain

$$\operatorname{g}(\bar{\nabla}\_{PY} QX, PZ) = PY \operatorname{g}(QX, PZ) - \operatorname{g}(QX, \bar{\nabla}\_{PY} PZ) = 0,\tag{69}$$

which implies that

$$P\tilde{\nabla}\_{PY}QX = 0 \text{ and } Q\tilde{\nabla}\_{PY}QX = \tilde{\nabla}\_{PY}QX. \tag{70}$$

From Eqs. (66), (68) and (70), it follows that

$$P\tilde{\nabla}\_Y X = \tilde{\nabla}\_{PY} PX + \tilde{\nabla}\_{\tilde{Q}Y} PX. \tag{71}$$

By using Eqs. (64) and (71), we obtain

$$\hat{\nabla}\left(\hat{\nabla}\_{Y}P\right)X = \hat{\nabla}\_{Y}PX - P\hat{\nabla}\_{Y}X = \hat{\nabla}\_{PY}PX + \hat{\nabla}\_{QY}PX - \hat{\nabla}\_{PY}PX - \hat{\nabla}\_{QY}PX = 0. \tag{72}$$

In the same way, we can find <sup>∇</sup><sup>~</sup> <sup>Q</sup> <sup>¼</sup> 0. Thus, we obtain

$$
\vec{\nabla}F = \vec{\nabla}(P \neg Q) = 0.\tag{73}
$$

This proves our assertion [2].

Theorem 2.3. Both distributions D and D<sup>⊥</sup> are parallel with respect to Levi-Civita connection ∇ if and only if they are integrable and their leaves are totally geodesic in M~ .

Proof: Let us assume both distributions D and D<sup>⊥</sup> are parallel. Since ∇ is a torsion free linear connection, we have

$$\mathbb{E}\left[X,Y\right] = \nabla\_X Y - \nabla\_Y X \in \Gamma(\mathcal{D}), \text{for any } X, Y \in \Gamma(\mathcal{D}) \tag{74}$$

and

$$\mathbb{E}\left[U,V\right] = \nabla\_U V - \nabla\_V U \in \Gamma(\mathcal{D}^\perp), \text{for any } U, V \in \Gamma(\mathcal{D}^\perp) \tag{75}$$

Thus, D and D<sup>⊥</sup> are integrable distributions. Now, let M be a leaf of D and denote by h the second fundamental form of the immersion of M in M~ . Then by the Gauss formula, we have

Sub-Manifolds of a Riemannian Manifold http://dx.doi.org/10.5772/65948 59

$$
\nabla\_X Y = \nabla\_X' Y + h(X, Y) \tag{76}
$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðDÞ, where <sup>∇</sup><sup>0</sup> denote the Levi-Civita connection on M. Since D is parallel from Eq. (76) we conclude <sup>h</sup> <sup>¼</sup> 0, that is, <sup>M</sup> is totally in <sup>M</sup><sup>~</sup> . In the same way, it follows that each leaf of D<sup>⊥</sup> is totally geodesic in M~ .

Conversely, suppose D and D<sup>⊥</sup> be integrable and their leaves are totally geodesic in M~ . Then by using Eq. (4), we have

$$\nabla\_X \mathbf{Y} \in \Gamma(\mathcal{D}) \text{ for any } X, Y \in \Gamma(\mathcal{D}) \tag{77}$$

and

<sup>∇</sup><sup>~</sup> YX <sup>¼</sup> <sup>∇</sup><sup>~</sup> PYPX <sup>þ</sup> <sup>∇</sup><sup>~</sup> PYQX <sup>þ</sup> <sup>∇</sup><sup>~</sup> QYPX <sup>þ</sup> <sup>∇</sup><sup>~</sup> QYQX, (66)

<sup>g</sup>ð∇<sup>~</sup> PYQX, PZÞ ¼ PYgðQX, PZÞ−gðQX, <sup>∇</sup><sup>~</sup> PYPZÞ ¼ <sup>0</sup>, (69)

<sup>ð</sup>∇<sup>~</sup> YPÞ<sup>X</sup> <sup>¼</sup> <sup>∇</sup><sup>~</sup> YPX−P∇<sup>~</sup> YX <sup>¼</sup> <sup>∇</sup><sup>~</sup> PYPX <sup>þ</sup> <sup>∇</sup><sup>~</sup> QYPX−∇<sup>~</sup> PYPX−∇<sup>~</sup> QYPX <sup>¼</sup> <sup>0</sup>: (72)

Theorem 2.3. Both distributions D and D<sup>⊥</sup> are parallel with respect to Levi-Civita connection ∇

Proof: Let us assume both distributions D and D<sup>⊥</sup> are parallel. Since ∇ is a torsion free linear

Thus, D and D<sup>⊥</sup> are integrable distributions. Now, let M be a leaf of D and denote by h the second fundamental form of the immersion of M in M~ . Then by the Gauss formula, we have

if and only if they are integrable and their leaves are totally geodesic in M~ .

<sup>P</sup>∇<sup>~</sup> PYQX <sup>¼</sup> 0 and Q∇<sup>~</sup> PYQX <sup>¼</sup> <sup>∇</sup><sup>~</sup> PYQX: (70)

<sup>P</sup>∇<sup>~</sup> YX <sup>¼</sup> <sup>∇</sup><sup>~</sup> PYPX <sup>þ</sup> <sup>∇</sup><sup>~</sup> QYPX: (71)

<sup>∇</sup><sup>~</sup> <sup>F</sup> <sup>¼</sup> <sup>∇</sup><sup>~</sup> <sup>ð</sup>P−QÞ ¼ <sup>0</sup>: (73)

½X,Y� ¼ ∇XY−∇YX∈ΓðDÞ, for any X, Y∈ΓðDÞ (74)

<sup>½</sup>U, <sup>V</sup>� ¼ <sup>∇</sup>UV−∇VU∈ΓðD<sup>⊥</sup>Þ, for any <sup>U</sup>, <sup>V</sup>∈ΓðD<sup>⊥</sup><sup>Þ</sup> (75)

<sup>Q</sup>∇<sup>~</sup> QYPX <sup>¼</sup> <sup>0</sup>: (68)

<sup>g</sup>ð∇<sup>~</sup> QYPX, QZÞ ¼ QYgðPX, QZÞ−gð∇QYQZ, PXÞ ¼ <sup>0</sup>−gð∇<sup>~</sup> QYQZ, PXÞ ¼ <sup>0</sup>, (67)

from which

58 Manifolds - Current Research Areas

In the same way, we obtain

which implies that

that is, <sup>∇</sup>QYPX∈ΓðD<sup>Þ</sup> and so <sup>P</sup>∇<sup>~</sup> QYPX <sup>¼</sup> <sup>∇</sup><sup>~</sup> QYPX,

From Eqs. (66), (68) and (70), it follows that

In the same way, we can find <sup>∇</sup><sup>~</sup> <sup>Q</sup> <sup>¼</sup> 0. Thus, we obtain

By using Eqs. (64) and (71), we obtain

This proves our assertion [2].

connection, we have

and

$$\nabla\_{\mathcal{U}} V \in \Gamma(\mathcal{D}^{\perp}) \text{ for any } \mathcal{U}, V \in \Gamma(\mathcal{D}^{\perp}). \tag{78}$$

Since g is a Riemannian metric tensor, we obtain

$$\lg(\nabla\_{\mathcal{U}}Y, V) = -\lg(Y, \nabla\_{\mathcal{U}}V) = 0\tag{79}$$

and

$$\lg(\nabla\_X V, Y) = -\lg(V, \nabla\_X Y) = 0\tag{80}$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðD<sup>Þ</sup> and <sup>U</sup>, <sup>V</sup>∈ΓðD<sup>⊥</sup>Þ: Thus, both distributions <sup>D</sup> and <sup>D</sup><sup>⊥</sup> are parallel on <sup>M</sup><sup>~</sup> .
