Open access peer-reviewed chapter

# Sub-Manifolds of a Riemannian Manifold

Written By

Mehmet Atçeken, Ümit Yıldırım and Süleyman Dirik

Submitted: 21 August 2016 Reviewed: 23 September 2016 Published: 18 January 2017

DOI: 10.5772/65948

From the Edited Volume

## Manifolds - Current Research Areas

Edited by Paul Bracken

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## Abstract

In this chapter, we introduce the theory of sub-manifolds of a Riemannian manifold. The fundamental notations are given. The theory of sub-manifolds of an almost Riemannian product manifold is one of the most interesting topics in differential geometry. According to the behaviour of the tangent bundle of a sub-manifold, with respect to the action of almost Riemannian product structure of the ambient manifolds, we have three typical classes of sub-manifolds such as invariant sub-manifolds, anti-invariant sub-manifolds and semi-invariant sub-manifolds. In addition, slant, semi-slant and pseudo-slant sub-manifolds are introduced by many geometers.

### Keywords

• Riemannian product manifold
• Riemannian product structure
• integral manifold
• a distribution on a manifold
• real product space forms
• a slant distribution

## 1. Introduction

Let i : M M ˜ be an immersion of an n-dimensional manifold M into an m-dimensional Riemannian manifold ( M ˜ , g ˜ ) . Denote by g = i * g ˜ the induced Riemannian metric on M . Thus, i become an isometric immersion and M is also a Riemannian manifold with the Riemannian metric g ( X , Y ) = g ˜ ( X , Y ) for any vector fields X , Y in M . The Riemannian metric g on M is called the induced metric on M . In local components, g i j = g A B B j B B i A with g = g j i d x j d x j and g ˜ = g B A d U B d U A .

If a vector field ξ p of M ˜ at a point p M satisfies

g ˜ ( X p , ξ p ) = 0 E1

for any vector X p of M at p , then ξ p 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].

By T 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 T M of M and the normal bundle T M of M in M ˜ , i . e . ,

T M ˜ = T M T M . E2

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 , ξ ) = 0 , g ( ξ , ξ ) = 1 , E3

where X is any arbitrary vector field on M .

By ˜ , denote the Riemannian connection on M ˜ and we put

˜ X Y = X Y + h ( X , Y ) E4

for any vector fields X , Y tangent to M , where X Y and h ( X , Y ) are tangential and the normal components of ˜ X Y , respectively. Formula ( 4 ) is called the Gauss formula for the sub-manifold M of a Riemannian manifold ( M ˜ , g ˜ ) .

Proposition 1.1. is the Riemannian connection of the induced metric g = i * g ˜ on M and h ( X , Y ) 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

˜ αX ( βY )=α{ X( β )Y+β ˜ X Y} =α{ X( β )Y+β X Y+βh( X,Y )}   αX βY+h( αX,βY ) =αβ X Y+αX( β )Y+αβh( X,Y ) E5

This implies that

α X ( β Y ) = α X ( β ) Y + α β X Y E6

and

h ( α X , β Y ) = α β h ( X , Y ) . E7

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 = ˜ X Y ˜ Y X [ X , Y ] = X Y + h ( X , Y ) X Y h ( Y , X ) [ X , Y ] . E8

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

X Y Y X = [ X , Y ] E9

and

h ( X , Y ) = h ( Y , X ) . E10

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

(Xg)(Y,Z)=(˜Xg˜)(Y,Z)=g˜(˜XY,Z)+g˜(Y,˜XZ)=g˜(XY+h(X,Y),Z)+g˜(Y,XZ+h(X,Z))=g˜(XY,Z)+g˜(Y,XZ)=g(XY,Z)+g(Y,XZ) E11

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 :   Γ ( T M ) × Γ ( T M ) Γ ( T M ) . E12

If h = 0 identically, then sub-manifold M is said to be totally geodesic, where Γ ( T M ) 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 ( M ˜ , g ˜ ) . By h , we denote the second fundamental form of M in M ˜ .

H = 1 n trace ( h ) is called the mean curvature vector of M in M ˜ . If H = 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

g ˜ ( h ( X , Y ) , H ) = λ g ( X , Y ) E13

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

h ( X , Y ) = g ( X , Y ) H . E14

It is clear that every minimal sub-manifold is pseudo-umbilical with λ = 0 . On the other hand, by a direct calculation, we can find λ = g ˜ ( H , H ) 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 M be a sub-manifold of a Riemannian manifold ( M ˜ , g ˜ ) and V be a normal vector field on M , X be a vector field on M . Then, we decompose

˜ X V = A V X + X V , E15

where A V X and X V denote the tangential and the normal components of X V , respectively. We can easily see that A V X and X V are both differentiable vector fields on M and normal bundle of M , respectively. Moreover, Eq. (15) is also called Weingarten formula.

Proposition 1.2. Let M be a sub-manifold of a Riemannian manifold ( M ˜ , g ˜ ) . Then

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

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

g ( A V X , Y ) = g ( h ( X , Y ) , V ) . E16

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

˜ αX ( βV )=α ˜ X ( βV ) =α{ X( β )V+β ˜ X V} A βV αX+ αX βV =αX( β )Vαβ A V X+αβ X V. E17

This implies that

A β V α X = α β A V X E18

and

α X β V = α X ( β ) V + α β X V . E19

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

X g ˜ ( Y , V ) = 0 , E20

for any X , Y Γ ( T M ) and V Γ ( T M ) .

Eq. (12) implies that

g ˜ ( ˜ X Y , V ) + g ˜ ( Y , ˜ X V ) = 0 . E21

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

g ˜ ( h ( X , Y ) , V ) g ( A V X , Y ) = 0 . E22

The proof is completed [3].

Let M be a sub-manifold of a Riemannian manifold ( M ˜ , g ˜ ) , and h and A V denote the second fundamental form and shape operator of M , respectively.

The covariant derivative of h and A V is, respectively, defined by

( ˜ X h ) ( Y , Z ) = X h ( Y , Z ) h ( X Y , Z ) h ( Y , X Z ) E23

and

( X A ) V Y = X ( A V Y ) A X V Y A V X Y E24

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

g ( ( X h ) ( Y , Z ) , V ) = g ( ( X A ) V Y , Z ) . E25

Example 1.1. We consider the isometric immersion

ϕ :   R 2 R 4 , E26
ϕ ( x 1 , x 2 ) = ( x 1 , x 1 2 1 , x 2 , x 2 2 1 ) E27

we note that M = ϕ ( R 2 ) R 4 is a two-dimensional sub-manifold of R 4 and the tangent bundle is spanned by the vectors

T M = S p { e 1 = ( x 1 2 1 , x 1 , 0 , 0 ) ,   e 2 = ( 0 , 0 , x 2 2 1 , x 2 ) } and the normal vector fields

T M = s p { w 1 = ( x 1 , x 1 2 1 , 0 , 0 ) , w 2 = ( 0 , 0 , x 1 , x 2 2 1 )   } . E28

By ˜ , we denote the Levi-Civita connection of R 4 , the coefficients of connection, are given by

˜ e 1   e 1 = 2 x 1 x 1 2 1 2 x 1 2 1 e 1 1 2 x 1 2 1 w 1 , E29
˜ e 2   e 2 = 2 x 2 x 2 2 1 2 x 2 2 1 e 2 1 2 x 2 2 1 w 2 E30

and

e 2   e 1 = 0. E31

Thus, we have h ( e 1 , e 1 ) = 1 2 x 1 2 1 w 1 , h ( e 2 , e 2 ) = 1 2 x 2 2 1 w 2 and h ( e 2 , e 1 ) = 0. The mean curvature vector of M = ϕ ( R 2 ) is given by

H = 1 2 ( w 1 + w 2 ) . E32

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

g ( A w 1 e 1 , e 1 ) = g ( h ( e 1 , e 1 ) , w 1 ) = 1 2 x 1 2 1 ( x 1 2 + x 1 2 1 ) = 1 , g ( A w 1 e 2 , e 2 ) = g ( h ( e 2 , e 2 ) , w 1 ) = 1 2 x 2 2 1 g ( w 1 , w 2 ) = 0 , g ( A w 1 e 1 , e 2 ) = 0 , E33

and

g ( A w 2 e 1 , e 1 ) = g ( h ( e 1 , e 1 ) , w 2 ) = 0 , g ( A w 2 e 1 , e 2 ) = 0 ,   g ( A w 2 e 2 , e 2 ) = 1. E34

Thus, we have

A w 1 = ( 1 0 0 0 )   and   A w 2 = ( 0 0 0 1 ) . E35

Now, let M be a sub-manifold of a Riemannian manifold ( M ˜ , g ) , R ˜ and R be the Riemannian curvature tensors of M ˜ and M , respectively. From then the Gauss and Weingarten formulas, we have

R˜(X,Y)Z=˜X˜YZ˜Y˜XZ˜[X,Y]Z=˜X(YZ+h(Y,Z))˜Y(XZ+h(X,Z))[X,Y]Zh([X,Y],Z)=˜XYZ+˜Xh(Y,Z)˜YXZ˜Yh(X,Z)[X,Y]Zh(XY,Z)+h(YX,Z)=XYZYXZ+h(X,YZ)h(XZ,Y)+Xh(Y,Z)Ah(Y,Z)XYh(X,Z)+Ah(X,Z)Y[X,Y]Zh(XY,Z)+h(YX,Z)=XYZYXZ[X,Y]Z+Xh(Y,Z)h(XY,Z)h(Y,XZ)Yh(X,Z)+h(YX,Z)+h(YZ,X)+Ah(X,Z)YAh(Y,Z)X=R(X,Y)Z+(Xh)(Y,Z)(Yh)(X,Z)+Ah(X,Z)YAh(Y,Z)X E36

from which

R ˜ ( X , Y ) Z = R ( X , Y ) Z + A h ( X , Z ) Y A h ( Y , Z ) X + ( X h ) ( Y , Z ) ( Y h ) ( X , Z ) , E37

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

g ( R ˜ ( X , Y ) Z , W ) = g ( R ( X , Y ) Z , W ) + g ( h ( Y , W ) , h ( X , Z ) ) g ( h ( Y , Z ) , h ( X , W ) ) . E38

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

( R ˜ ( X , Y ) Z ) = ( X h ) ( Y , Z ) ( Y h ) ( X , Z ) . E39

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

In particular, if M ˜ is of constant curvature, R ˜ ( X , Y ) Z is tangent to M , 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 of the normal bundle of the sub-manifold M by

R ( X , Y ) V = X Y V Y X V [ X , Y ] V E40

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

R˜(X,Y)Z=˜X˜YZ˜Y˜XZ˜[X,Y]Z=˜X(YZ+h(Y,Z))˜Y(XZ+h(X,Z))[X,Y]Zh([X,Y],Z)=˜XYZ+˜Xh(Y,Z)˜YXZ˜Yh(X,Z)[X,Y]Zh(XY,Z)+h(YX,Z)=XYZYXZ+h(X,YZ)h(XZ,Y)+Xh(Y,Z)Ah(Y,Z)XYh(X,Z)+Ah(X,Z)Y[X,Y]Zh(XY,Z)+h(YX,Z)=XYZYXZ[X,Y]Z+Xh(Y,Z)h(XY,Z)h(Y,XZ)Yh(X,Z)+h(YX,Z)+h(YZ,X)+Ah(X,Z)YAh(Y,Z)X=R(X,Y)Z+(Xh)(Y,Z)(Yh)(X,Z)+Ah(X,Z)YAh(Y,Z)X E41

For any normal vector U to M , we obtain

g( R˜(X,Y)V,U)=g(R(X,Y)V,U)+g(h(AVX,Y),U)g(h(X,AVY),U)=g(R(X,Y)V,U)+g(AUY,AVX)g(AVY,AUX)=g(R(X,Y)V,U)+g(AVAUY,X)g(AUAVY,X) E42

Since [ A U , A V ] = A U A V A V A U , Eq. (42) implies

g (   R ˜ ( X , Y ) V , U ) =   g ( R ( X , Y ) V , U ) + g ( [ A U , A V ] Y , X ) . E43

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

If R = 0 , then the normal connection of M is said to be flat [2].

When ( R ˜ ( X , Y ) V ) = 0 , the normal connection of the sub-manifold M is flat if and only if the second fundamental form M is commutative, i.e. [ A U , A V ] = 0 for all U , V . If the ambient space M ˜ is real space form, then ( R ˜ ( X , Y ) V ) = 0 and hence the normal connection of M is flat if and only if the second fundamental form is commutative. If R ˜ ( X , Y ) Z tangent to M , then equation of codazzi Eq. (37) reduces to

( X h ) ( Y , Z ) = ( Y h ) ( X , Z ) E44

which is equivalent to

( X A ) V Y = ( Y A ) V X . E45

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

R ˜ ( X , Y ) Z = c { g ( Y , Z ) X g ( X , Z ) Y } E46

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

Since R ˜ ( X , Y ) Z is tangent to M , the equation of Gauss and the equation of Ricci reduce to

g(R(X,Y)Z,W)=c{g(Y,Z)g(X,W)g(X,Z)g(Y,W)}+g(h(Y,Z),h(X,W))g(h(Y,W),h(X,Z)) E47

and

g ( R ( X , Y ) V , U ) = g ( [ A U , A V ] X , Y ) , E48

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

g(R(X,Y)Z,W)=c{g(Y,Z)g(X,W)g(X,Z)g(Y,W)}+g(H,H){g(Y,Z)g(X,W)g(X,Z)g(Y,W)}={c+g(H,H)}{g(Y,Z)g(X,W)g(X,Z)g(Y,W)} E49

This shows that the sub-manifold M is of constant curvature c + H 2 for n > 2 . If n = 2 , H = constant follows from the equation of Codazzi [3].

This proves the proposition.

On the other hand, for any orthonormal basis { e a } of normal space, we have

g(Y,Z)g(X,W)g(X,Z)g(Y,W)=a[g(h(Y,Z),ea)g(h(X,W),ea) g(h(X,Z),ea)g(h(Y,W),ea)]=ag(AeaY,Z)g(AeaX,W)g(AeaX,Z)g(AeaY,W) E50

Thus, Eq. (45) can be rewritten as

g(R(X,Y)Z,W)=c{g(Y,Z)g(X,W)g(X,Z)g(Y,W)}+a[g(AeaY,Z)g(AeaX,W)g(AeaX,Z)g(AeaY,W)] E51

By using A e a , 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 ) = c { n g ( X , Y ) g ( e i , X ) g ( e i , Y ) } E52
S( X,Y )=c{ ng( X,Y )g( e i ,X )g( e i ,Y )} e a [ g( A e a e i , e i )g( A e a X,Y )g( A e a X, e i )g( A e a e i ,Y )] =c( n1 )g( X,Y )+ e a [ Tr( A e a )g( A e a X,Y )g( A e a X, A e a Y )], E53

where { e 1 , e 2 , , e n } are orthonormal basis of M .

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

r = c n ( n 1 ) e a T r 2 ( A e a ) e a T r ( A e a ) 2 E54

e a T r ( A e a ) 2 is the square of the length of the second fundamental form of M , which is denoted by | A e a | 2 . Thus, we also have

h 2 = i , j = 1 n g ( h ( e i , e j ) , h ( e i , e j ) ) = A 2 . E55

## 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 p of M ˜ an m-dimensional linear subspace D p of T M ˜ ( p ) . A vector field X on M ˜ belongs to D if we have X p   D p for each p M ˜ . When this happens, we write X Γ ( D ) . The distribution D is said to be differentiable if for any p M ˜ , there exist m-differentiable linearly independent vector fields X j Γ ( D ) in a neighbordhood of p.

The distribution D is said to be involutive if for all vector fields X , Y Γ ( D ) we have [ X , Y ] Γ ( D ) . A sub-manifold M of M ˜ is said to be an integral manifold of D if for every point p   M , D p 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

˜ X Y Γ ( D ) for all  X Γ ( T M ˜ )   and   Y Γ ( D ) E56

Now, let ( M ˜ , g ˜ ) be Riemannian manifold and D be a distribution on M ˜ . We suppose M ˜ is endowed with two complementary distribution D and D , i . e . , we have T M ˜ = D D . Denoted by P and Q the projections of T M ˜ to D and D , respectively.

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

X Y = P X ' P Y + Q X ' Q Y + P S ( X , P Y ) + Q S ( X , Q Y ) E57

for any X , Y Γ ( T M ˜ ) , where ' and S are, respectively, an arbitrary linear connection and arbitrary tensor field of type ( 1 ,   2 ) on M ˜ .

Proof: Suppose ' is an arbitrary linear connection on M ˜ . Then, any linear connection on M ˜ is given by

X Y = X ' Y + S ( X , Y ) E58

for any X , Y Γ ( T M ˜ ) . We can put

X = P X + Q X E59

for any X Γ ( T M ˜ ) . Then, we have

XY=X(PY+QY)=XPY+XQY=X'PY+S(X,PY)+X'QY+S(X,QY)=PX'PY+QX'PY+PS(X,PY)+QS(X,PY)+PX'QY+QX'QY+PS(X,QY)+QS(X,QY) E60

for any X , Y Γ ( T M ˜ ) .

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

ϕ ( X P Y ) = 0   and P ( X Q Y ) = 0.   E61

From Eqs. (58) and (61), it follows that D and D are parallel with respect to if and only if

Q X ' P Y + Q S ( X , P Y ) = 0   and P X ' Q Y + P S ( X , Q Y ) = 0.   E62

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

Next, by means of the projections P and Q , we define a tensor field F of type ( 1 ,   1 ) on M ˜ by

F X = P X Q X E63

for any X Γ ( T M ˜ ) . By a direct calculation, it follows that F 2 = I . Thus, we say that F defines an almost product structure on M ˜ . The covariant derivative of F is defined by

( X F ) Y = X F Y F X Y E64

for all X , Y Γ ( T M ˜ ) . We say that the almost product structure F is parallel with respect to the connection , if we have X F = 0 . In this case, F is called the Riemannian product structure [2].

Theorem 2.2. Let ( M ˜ , g ˜ ) be a Riemannian manifold and D , D be orthogonal distributions on M ˜ such that T M ˜ = D D . Both distributions D and D are parallel with respect to if and only if F is a Riemannian product structure.

Proof: For any X , Y Γ ( T M ˜ ) , we can write

˜ Y P X =   ˜ P Y P X + ˜ Q Y P X E65

and

˜ Y X = ˜ P Y P X + ˜ P Y Q X + ˜ Q Y P X + ˜ Q Y Q X ,   E66

from which

g ( ˜ Q Y P X , Q Z ) = Q Y g ( P X , Q Z ) g ( Q Y Q Z , P X ) = 0 g ( ˜ Q Y Q Z , P X ) = 0 , E67

that is, Q Y P X Γ ( D ) and so P ˜ Q Y P X = ˜ Q Y P X ,

Q ˜ Q Y P X = 0.   E68

In the same way, we obtain

g ( ˜ P Y Q X , P Z ) = P Y g ( Q X , P Z ) g ( Q X , ˜ P Y P Z ) = 0 , E69

which implies that

P ˜ P Y Q X = 0   and Q ˜ P Y Q X = ˜ P Y Q X .   E70

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

P ˜ Y X = ˜ P Y P X + ˜ Q Y P X .   E71

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

( ˜   Y P ) X = ˜   Y P X P ˜ Y X   = ˜ P Y P X + ˜ Q Y P X ˜ P Y P X ˜ Q Y P X = 0 . E72

In the same way, we can find ˜ Q = 0 . Thus, we obtain

˜ F = ˜ ( P Q ) = 0. E73

This proves our assertion [2].

Theorem 2.3. Both distributions D and D 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 are parallel. Since is a torsion free linear connection, we have

[ X , Y ] = X Y Y X Γ ( D ) , for any X , Y Γ ( D ) E74

and

[ U , V ] = U V V U Γ ( D ) , for any U , V Γ ( D ) E75

Thus, D and D 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

X Y = X ' Y + h ( X , Y ) E76

for any X , Y Γ ( D ) , where ' denote the Levi-Civita connection on M . Since D is parallel from Eq. (76) we conclude h = 0 , that is, M is totally in M ˜ . In the same way, it follows that each leaf of D is totally geodesic in M ˜ .

Conversely, suppose D and D be integrable and their leaves are totally geodesic in M ˜ . Then by using Eq. (4), we have

X Y Γ ( D )   for any  X , Y Γ ( D ) E77

and

U V Γ ( D )   for any  U , V Γ ( D ) . E78

Since g is a Riemannian metric tensor, we obtain

g ( U Y , V ) = g ( Y , U V ) = 0 E79

and

g ( X V , Y ) = g ( V , X Y ) = 0 E80

for any X , Y Γ ( D ) and U , V Γ ( D ) . Thus, both distributions D and D are parallel on M ˜ .

## 3. Locally decomposable Riemannian manifolds

Let ( M ˜ , g ˜ ) be n dimensional Riemannian manifold and F be a tensor ( 1 , 1 ) type on M ˜ such that F 2 = I , F I .

If the Riemannian metric tensor g ˜ satisfying

g ˜ ( X , Y ) = g ˜ ( F X , F Y ) E81

for any X , Y Γ ( T M ˜ ) then M ˜ is called almost Riemannian product manifold and F is said to be almost Riemannian product structure. If F is parallel, that is, (   ˜ X F ) Y = 0 , then M ˜ is said to be locally decomposable Riemannian manifold.

Now, let M ˜ be an almost Riemannian product manifold. We put

P = 1 2 ( I + F ) , Q = 1 2 ( I F ) . E82

Then, we have

P + Q = I ,   P 2 = P ,   Q 2 = Q ,   P Q = Q P = 0   and  F = P Q . E83

Thus, P and Q define two complementary distributions P and Q globally. Since F 2 = I , we easily see that the eigenvalues of F are 1 and 1 . An eigenvector corresponding to the eigenvalue 1 is in P and an eigenvector corresponding to 1 is in Q . If F has eigenvalue 1 of multiplicity P and eigenvalue 1 of multiplicity q , then the dimension of P is p and that of Q is q . Conversely, if there exist in M ˜ two globally complementary distributions P and Q of dimension p and q , respectively. Then, we can define an almost Riemannian product structure F on M ˜ by M ˜ by F = P Q [7].

Let ( M ˜ , g ˜ , F ) be a locally decomposable Riemannian manifold and we denote the integral manifolds of the distributions P and Q by M p and M q , respectively. Then we can write M ˜ = M p X M q , ( p , q > 2 ) . Also, we denote the components of the Riemannian curvature R of M ˜ by R d c b a , 1 a , b , c , d n = p + q .

Now, we suppose that the two components are both of constant curvature λ and μ . Then, we have

R d c b a = λ { g d a g c b g c a g d b } E84

and

R z y x w = μ { g z w g y x g y w g z x } . E85

Then, the above equations may also be written in the form

R k j i h = 1 4 ( λ + μ ) { ( g k h g j i g j h g k i ) + ( F k h F j i F j h F k i ) } + 1 4 ( λ μ ) { ( F k h g j i F j h g k i ) + ( g k h F j i g j h F k i ) } . E86

Conversely, suppose that the curvature tensor of a locally decomposable Riemannian manifold has the form

R kjih = 1 4 ( λ+μ ){ ( g kh g ji g jh g ki )+( F kh F ji F jh F ki )} + 1 4 ( λμ ){ ( F kh g ji F jh g ki )+( g kh F ji g jh F ki )}. E87

Then, we have

R c d b a = 2 ( a + b ) { g d a g c b g c a g d b } E88

and

R z y x w = 2 ( a b ) { g z w g y x g y w g z x } . E89

Let M ˜ be an m dimensional almost Riemannian product manifold with the Riemannian structure ( F , g ˜ ) and M be an n dimensional sub-manifold of M ˜ . For any vector field X tangent to M , we put

F X = f X + w X , E90

where f X and w X denote the tangential and normal components of F X , with respect to M , respectively. In the same way, for V Γ ( T M ) , we also put

F V = B V + C V , E91

where B V and C V denote the tangential and normal components of F V , respectively.

Then, we have

f 2 + B w = I , C w + w f = 0   E92

and

f B + B C = 0 ,   w B + C 2 = I . E93

On the other hand, we can easily see that

g ( X , f Y ) = g ( f X , Y ) E94

and

g ( X , Y ) = g ( f X , f Y ) + g ( w X , w Y ) E95

for any X , Y Γ ( T M ) [6].

If w X = 0 for all X Γ ( T M ) , then M is said to be invariant sub-manifold in M ˜ , i . e . ,   F ( T M ( p ) ) T M ( p ) for each p M . In this case, f 2 = I and g ( f X , f Y ) = g ( X , Y ) . Thus, ( f , g ) defines an almost product Riemannian on M .

Conversely, ( f , g ) is an almost product Riemannian structure on M , the w = 0 and hence M is an invariant sub-manifold in M ˜ .

Consequently, we can give the following theorem [7].

Theorem 3.1. Let M be a sub-manifold of an almost Riemannian product manifold M ˜ with almost Riemannian product structure ( F , g ˜ ) . The induced structure ( f , g ) on M is an almost Riemannian product structure if and only if M is an invariant sub-manifold of M ˜ .

Definition 3.1. Let M be a sub-manifold of an almost Riemannian product M ˜ with almost product Riemannian structure ( F , g ˜ ) . For each non-zero vector X p T M ( p ) at p M , we denote the slant angle between F X p and T M ( p ) by θ ( p ) . Then M said to be slant sub-manifold if the angle θ ( p ) is constant, i . e . , it is independent of the choice of p M and X p T M ( p ) [5].

Thus, invariant and anti-invariant immersions are slant immersions with slant angle θ = 0 and θ = π 2 , respectively. A proper slant immersion is neither invariant nor anti-invariant.

Theorem 3.2. Let M be a sub-manifold of an almost Riemannian product manifold M ˜ with almost product Riemannian structure ( F , g ˜ ) . M is a slant sub-manifold if and only if there exists a constant λ ( 0 , 1 ) , such tha

f 2 = λ I . E96

Furthermore, if the slant angle is θ , then it satisfies λ = cos 2 θ [9].

Definition 3.2. Let M be a sub-manifold of an almost Riemannian product manifold M ˜ with almost Riemannian product structure ( F , g ˜ ) . M is said to be semi-slant sub-manifold if there exist distributions D θ and D T on M such that

(i) T M has the orthogonal direct decomposition T M = D D T .

(ii) The distribution D θ is a slant distribution with slant angle θ .

(iii) The distribution D T is an invariant distribution, . e . , F ( D T ) D T .

In a semi-slant sub-manifold, if θ = π 2 , then semi-slant sub-manifold is called semi-invariant sub-manifold [8].

Example 3.1. Now, let us consider an immersed sub-manifold M in R 7 given by the equations

x 1 2 + x 2 2 = x 5 2 + x 6 2 ,   x 3 + x 4 = 0. E97

By direct calculations, it is easy to check that the tangent bundle of M is spanned by the vectors

z 1 = cos θ x 1 + sin θ x 2 + cos β x 5 + sin β x 6 z 2 = u sin θ x 1 + u cos θ x 2 ,   z 3 =   x 3 x 4 , z 4 = u sin β x 5 + u cos β x 6 ,   z 5 =   x 7 , E98

where θ , β and u denote arbitrary parameters.

For the coordinate system of R 7 = { ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) | x i R ,   1 i 7 } , we define the almost product Riemannian structure F as follows:

F ( x i ) = x i ,   F ( x j ) = x j ,   1 i 3   and   4 j 7. E99

Since F z 1 and F z 3 are orthogonal to M and F z 2 ,   F z 4 ,   F z 5 are tangent to M , we can choose a D = S p { z 2 , z 4 , z 5 } and D = S p { z 1 , z 3 } . Thus, M is a 5 dimensional semi-invariant sub-manifold of R 7 with usual almost Riemannian product structure ( F , < , > ) .

Example 3.2. Let M be sub-manifold of R 8 by given

( u + v , u v , u cos α , u sin α , u + v , u v , u cos β , u sin β ) E100

where u , v and β are the arbitrary parameters. By direct calculations, we can easily see that the tangent bundle of M is spanned by

e 1 = x 1 + x 2 + cos α x 3 + sin α x 4 + x 5 x 6 + cos β x 7 + sin β x 8 e 2 = x 1 x 2 + x 5 + x 6 ,   e 3 = u sin x 3 + u cos α x 4 , e 4 = u sin β x 7 + u cos β x 8 . E101

For the almost Riemannian product structure F of R 8 = R 4 x R 4 , F ( T M ) is spanned by vectors

F e 1 = x 1 + x 2 + cos α x 3 + sin α x 4 x 5 + x 6 cos β x 7 sin β x 8 , F e 2 = x 1 x 2 x 5 x 6 ,   . F e 3 = e 3 and F e 4 = e 4 . E102

Since F e 1 and F e 2 are orthogonal to M and F e 3 and F e 4 are tangent to M , we can choose D T = S p { e 3 , e 4 } and D = S p { e 1 , e 2 } . Thus, M is a four-dimensional semi-invariant sub-manifold of R 8 = R 4 x R 4 with usual Riemannian product structure F .

Definition 3.3. Let M be a sub-manifold of an almost Riemannian product manifold M ˜ with almost Riemannian product structure ( F , g ˜ ) . M is said to be pseudo-slant sub-manifold if there exist distributions D θ and D on M such that

1. The tangent bundle T M = D θ D .

2. The distribution D θ is a slant distribution with slant angle θ .

3. The distribution D is an anti-invariant distribution, i . e . , F ( D ) T M .

As a special case, if θ = 0 and θ = π 2 , then pseudo-slant sub-manifold becomes semi-invariant and anti-invariant sub-manifolds, respectively.

Example 3.3. Let M be a sub-manifold of R 6 by the given equation

( 3 u , v , v sin θ , v cos θ , s cos t , s cos t ) E103

where u , v , s and t arbitrary parameters and θ is a constant.

We can check that the tangent bundle of M is spanned by the tangent vectors

e 1 = 3 x 1 ,   e 2 = y 1 + sin θ x 2 + cos θ y 2 , e 3 = cos t x 3 cos t y 3 ,   e 4 = s sin t x 3 + s sin t y 3 . E104

For the almost product Riemannian structure F of R 6 whose coordinate systems ( x 1 , y 1 , x 2 , y 2 , x 3 , y 3 ) choosing

F ( x i ) = y i ,   1 i 3 , F ( y j ) = x j ,   1 j 3 , E105

Then, we have

F e 1 = 3 y 1 ,   F e 2 = x 1 + sin θ y 2 cos θ x 2 F e 3 = cos t y 3 + cos t x 3 ,   F e 4 = s sin t y 3 s sin t x 3 . E106

Thus, D θ = S p { e 1 , e 2 } is a slant distribution with slant angle α = π 4 . Since F e 3 and F e 4 are orthogonal to M , D = S p { e 3 , e 4 } is an anti-invariant distribution, that is, M is a 4-dimensional proper pseudo-slant sub-manifold of R 6 with its almost Riemannian product structure ( F , < , > ) .

## References

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Written By

Mehmet Atçeken, Ümit Yıldırım and Süleyman Dirik

Submitted: 21 August 2016 Reviewed: 23 September 2016 Published: 18 January 2017