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Mathematics » "Manifolds - Current Research Areas", book edited by Paul Bracken, ISBN 978-953-51-2872-4, Print ISBN 978-953-51-2871-7, Published: January 18, 2017 under CC BY 3.0 license. © The Author(s).

Sub-Manifolds of a Riemannian Manifold

By Mehmet Atçeken, Ümit Yıldırım and Süleyman Dirik
DOI: 10.5772/65948

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Sub-Manifolds of a Riemannian Manifold

Mehmet Atçeken1, Ümit Yıldırım1 and Süleyman Dirik2
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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:MM˜ 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, gij=gABBjBBiA with g=gjidxjdxj and g˜=gBAdUBdUA .

If a vector field ξp of M˜ at a point pM satisfies

 g˜(Xp,ξp)=0 (1)

for any vector Xp of M at p , then ξp 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 TM , 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 TM of M in M˜ , i.e.,

 TM˜=TM⊕T⊥M. (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,ξ)=0,g(ξ,ξ)=1, (3)

where X is any arbitrary vector field on M .

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

 ∇˜XY=∇XY+h(X,Y) (4)

for any vector fields X,Y tangent to M , where XY and h(X,Y) are tangential and the normal components of ˜XY , 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∇˜XY}=α{X(β)Y+β∇XY+βh(X,Y)} ∇αXβY+h(αX,βY)=αβ∇XY+αX(β)Y+αβh(X,Y) (5)

This implies that

 ∇αX(βY)=αX(β)Y+αβ∇XY (6)

and

 h(αX,βY)=αβh(X,Y). (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=∇˜XY−∇˜YX−[X,Y]=∇XY+h(X,Y)−∇XY−h(Y,X)−[X,Y]. (8)

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

 ∇XY−∇YX=[X,Y] (9)

and

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

 (∇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) (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: Γ(TM)×Γ(TM)→Γ(T⊥M). (12)

If h=0 identically, then sub-manifold M is said to be totally geodesic, where Γ(TM) 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=1ntrace(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)=λg(X,Y) (13)

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

 h(X,Y)=g(X,Y)H. (14)

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

 ∇˜XV=−AVX+∇X⊥V, (15)

where AVX and XV denote the tangential and the normal components of XV , respectively. We can easily see that AVX and XV 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) AVX is bilinear in vector fields V and X . Hence, AVX at point pM depends only on vector fields Vp and Xp .

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

 g(AVX,Y)=g(h(X,Y),V). (16)

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

 ∇˜αX(βV)=α∇˜X(βV)=α{X(β)V+β∇˜XV}−AβVαX+∇αX⊥βV=αX(β)V−αβAVX+αβ∇X⊥V. (17)

This implies that

 AβVαX=αβAVX (18)

and

 ∇αX⊥βV=αX(β)V+αβ∇X⊥V. (19)

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

 Xg˜(Y,V)=0, (20)

for any X,YΓ(TM) and VΓ(TM) .

Eq. (12) implies that

 g˜(∇˜XY,V)+g˜(Y,∇˜XV)=0. (21)

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

 g˜(h(X,Y),V)−g(AVX,Y)=0. (22)

The proof is completed [3].

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

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

 (∇˜Xh)(Y,Z)=∇X⊥h(Y,Z)−h(∇XY,Z)−h(Y,∇XZ) (23)

and

 (∇XA)VY=∇X(AVY)−A∇X⊥VY−AV∇XY (24)

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

 g((∇Xh)(Y,Z),V)=g((∇XA)VY,Z). (25)

Example 1.1. We consider the isometric immersion

 ϕ: R2→R4, (26)
 ϕ(x1,x2)=(x1,x12−1,x2,x22−1) (27)

we note that M=ϕ(R2)R4 is a two-dimensional sub-manifold of R4 and the tangent bundle is spanned by the vectors

TM=Sp{e1=(x121,x1,0,0), e2=(0,0,x221,x2)} and the normal vector fields

 T⊥M=sp{w1=(−x1,x12−1,0,0),w2=(0,0,−x1,x22−1) }. (28)

By ˜ , we denote the Levi-Civita connection of R4 , the coefficients of connection, are given by

 ∇˜e1 e1=2x1x12−12x12−1e1−12x12−1w1, (29)
 ∇˜e2 e2=2x2x22−12x22−1e2−12x22−1w2 (30)

and

 ∇e2 e1=0. (31)

Thus, we have h(e1,e1)=12x121w1 , h(e2,e2)=12x221w2 and h(e2,e1)=0. The mean curvature vector of M=ϕ(R2) is given by

 H=−12(w1+w2). (32)

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

 g(Aw1e1,e1)=g(h(e1,e1),w1)=−12x12−1(x12+x12−1)=−1,g(Aw1e2,e2)=g(h(e2,e2),w1)=−12x22−1g(w1,w2)=0,g(Aw1e1,e2)=0, (33)

and

 g(Aw2e1,e1)=g(h(e1,e1),w2)=0,g(Aw2e1,e2)=0, g(Aw2e2,e2)=1. (34)

Thus, we have

 Aw1=(−1000) and Aw2=(000−1). (35)

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]Z−h([X,Y],Z)=∇˜X∇YZ+∇˜Xh(Y,Z)−∇˜Y∇XZ−∇˜Yh(X,Z)−∇[X,Y]Z−h(∇XY,Z)+h(∇YX,Z)=∇X∇YZ−∇Y∇XZ+h(X,∇YZ)−h(∇XZ,Y)+∇X⊥h(Y,Z)−Ah(Y,Z)X−∇Y⊥h(X,Z)+Ah(X,Z)Y−∇[X,Y]Z−h(∇XY,Z)+h(∇YX,Z)=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z+∇X⊥h(Y,Z)−h(∇XY,Z)−h(Y,∇XZ)−∇Y⊥h(X,Z)+h(∇YX,Z)+h(∇YZ,X)+Ah(X,Z)Y−Ah(Y,Z)X=R(X,Y)Z+(∇Xh)(Y,Z)−(∇Yh)(X,Z)+Ah(X,Z)Y−Ah(Y,Z)X (36)

from which

 R˜(X,Y)Z=R(X,Y)Z+Ah(X,Z)Y−Ah(Y,Z)X+(∇Xh)(Y,Z)−(∇Yh)(X,Z), (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

 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)). (38)

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

 (R˜(X,Y)Z)⊥=(∇Xh)(Y,Z)−(∇Yh)(X,Z). (39)

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 (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

 R˜(X,Y)Z=∇˜X∇˜YZ−∇˜Y∇˜XZ−∇˜[X,Y]Z=∇˜X(∇YZ+h(Y,Z))−∇˜Y(∇XZ+h(X,Z))−∇[X,Y]Z−h([X,Y],Z)=∇˜X∇YZ+∇˜Xh(Y,Z)−∇˜Y∇XZ−∇˜Yh(X,Z)−∇[X,Y]Z−h(∇XY,Z)+h(∇YX,Z)=∇X∇YZ−∇Y∇XZ+h(X,∇YZ)−h(∇XZ,Y)+∇X⊥h(Y,Z)−Ah(Y,Z)X−∇Y⊥h(X,Z)+Ah(X,Z)Y−∇[X,Y]Z−h(∇XY,Z)+h(∇YX,Z)=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z+∇X⊥h(Y,Z)−h(∇XY,Z)−h(Y,∇XZ)−∇Y⊥h(X,Z)+h(∇YX,Z)+h(∇YZ,X)+Ah(X,Z)Y−Ah(Y,Z)X=R(X,Y)Z+(∇Xh)(Y,Z)−(∇Yh)(X,Z)+Ah(X,Z)Y−Ah(Y,Z)X (41)

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) (42)

Since [AU,AV]=AUAVAVAU , Eq. (42) implies

 g( R˜(X,Y)V,U)= g(R⊥(X,Y)V,U)+g([AU,AV]Y,X). (43)

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. [AU,AV]=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

 (∇Xh)(Y,Z)=(∇Yh)(X,Z) (44)

which is equivalent to

 (∇XA)VY=(∇YA)VX. (45)

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} (46)

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)) (47)

and

 g(R⊥(X,Y)V,U)=g([AU,AV]X,Y), (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

 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)} (49)

This shows that the sub-manifold M is of constant curvature c+H2 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 {ea} 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) (50)

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)] (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)=c{ng(X,Y)−g(ei,X)g(ei,Y)} (52)
 S(X,Y)=c{ng(X,Y)−g(ei,X)g(ei,Y)}∑ea[g(Aeaei,ei)g(AeaX,Y)−g(AeaX,ei)g(Aeaei,Y)]=c(n−1)g(X,Y)+∑ea[Tr(Aea)g(AeaX,Y)−g(AeaX,AeaY)], (53)

where {e1,e2,,en} are orthonormal basis of M .

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

 r=cn(n−1)∑eaTr2(Aea)−∑eaTr(Aea)2 (54)

eaTr(Aea)2 is the square of the length of the second fundamental form of M , which is denoted by |Aea|2 . Thus, we also have

 ‖h2‖=∑i,j=1ng(h(ei,ej),h(ei,ej))=‖A2‖. (55)

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 Dp of TM˜(p) . A vector field X on M˜ belongs to D if we have Xp Dp for each pM˜ . When this happens, we write XΓ(D) . The distribution D is said to be differentiable if for any pM˜ , there exist m-differentiable linearly independent vector fields XjΓ(D) in a neighbordhood of p.

The distribution D is said to be involutive if for all vector fields X,YΓ(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 , Dp 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 pM˜ , 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

 ∇˜XY∈Γ(D)for all X∈Γ(TM˜) and Y∈Γ(D) (56)

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 TM˜=DD . Denoted by P and Q the projections of TM˜ 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

 ∇XY=P∇X'PY+Q∇X'QY+PS(X,PY)+QS(X,QY) (57)

for any X,YΓ(TM˜) , 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

 ∇XY=∇X'Y+S(X,Y) (58)

for any X,YΓ(TM˜) . We can put

 X=PX+QX (59)

for any XΓ(TM˜) . Then, we have

 ∇XY=∇X(PY+QY)=∇XPY+∇XQY=∇X'PY+S(X,PY)+∇X'QY+S(X,QY)=P∇X'PY+Q∇X'PY+PS(X,PY)+QS(X,PY)+P∇X'QY+Q∇X'QY+PS(X,QY)+QS(X,QY) (60)

for any X,YΓ(TM˜) .

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

 ϕ(∇XPY)=0 and P(∇XQY)=0. (61)

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

 Q∇X'PY+QS(X,PY)=0 and P∇X'QY+PS(X,QY)=0. (62)

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

 FX=PX−QX (63)

for any XΓ(TM˜) . By a direct calculation, it follows that F2=I . Thus, we say that F defines an almost product structure on M˜ . The covariant derivative of F is defined by

 (∇XF)Y=∇XFY−F∇XY (64)

for all X,YΓ(TM˜) . We say that the almost product structure F is parallel with respect to the connection , if we have XF=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 TM˜=DD. Both distributions D and D are parallel with respect to if and only if F is a Riemannian product structure.

Proof: For any X,YΓ(TM˜) , we can write

 ∇˜YPX= ∇˜PYPX+∇˜QYPX (65)

and

 ∇˜YX=∇˜PYPX+∇˜PYQX+∇˜QYPX+∇˜QYQX, (66)

from which

 g(∇˜QYPX,QZ)=QYg(PX,QZ)−g(∇QYQZ,PX)=0−g(∇˜QYQZ,PX)=0, (67)

that is, QYPXΓ(D) and so P˜QYPX=˜QYPX ,

 Q∇˜QYPX=0. (68)

In the same way, we obtain

 g(∇˜PYQX,PZ)=PYg(QX,PZ)−g(QX,∇˜PYPZ)=0, (69)

which implies that

 P∇˜PYQX=0 and Q∇˜PYQX=∇˜PYQX. (70)

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

 P∇˜YX=∇˜PYPX+∇˜QYPX. (71)

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

 (∇˜ YP)X=∇˜ YPX−P∇˜YX =∇˜PYPX+∇˜QYPX−∇˜PYPX−∇˜QYPX=0. (72)

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

 ∇˜F=∇˜(P−Q)=0. (73)

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]=∇XY−∇YX∈Γ(D),for anyX,Y∈Γ(D) (74)

and

 [U,V]=∇UV−∇VU∈Γ(D⊥),for anyU,V∈Γ(D⊥) (75)

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

 ∇XY=∇X'Y+h(X,Y) (76)

for any X,YΓ(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

 ∇XY∈Γ(D) for any X,Y∈Γ(D) (77)

and

 ∇UV∈Γ(D⊥) for any U,V∈Γ(D⊥). (78)

Since g is a Riemannian metric tensor, we obtain

 g(∇UY,V)=−g(Y,∇UV)=0 (79)

and

 g(∇XV,Y)=−g(V,∇XY)=0 (80)

for any X,YΓ(D) and U,VΓ(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 F2=I , FI .

If the Riemannian metric tensor g˜ satisfying

 g˜(X,Y)=g˜(FX,FY) (81)

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

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

 P=12(I+F),Q=12(I−F). (82)

Then, we have

 P+Q=I, P2=P, Q2=Q, PQ=QP=0 and F=P−Q. (83)

Thus, P and Q define two complementary distributions P and Q globally. Since F2=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=PQ [7].

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

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

 Rdcba=λ{gdagcb−gcagdb} (84)

and

 Rzyxw=μ{gzwgyx−gywgzx}. (85)

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

 Rkjih=14(λ+μ){(gkhgji−gjhgki)+(FkhFji−FjhFki)}+14(λ−μ){(Fkhgji−Fjhgki)+(gkhFji−gjhFki)}. (86)

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

 Rkjih=14(λ+μ){(gkhgji−gjhgki)+(FkhFji−FjhFki)}+14(λ−μ){(Fkhgji−Fjhgki)+(gkhFji−gjhFki)}. (87)

Then, we have

 Rcdba=2(a+b){gdagcb−gcagdb} (88)

and

 Rzyxw=2(a−b){gzwgyx−gywgzx}. (89)

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

 FX=fX+wX, (90)

where fX and wX denote the tangential and normal components of FX , with respect to M , respectively. In the same way, for VΓ(TM) , we also put

 FV=BV+CV, (91)

where BV and CV denote the tangential and normal components of FV , respectively.

Then, we have

 f2+Bw=I,Cw+wf=0 (92)

and

 fB+BC=0, wB+C2=I. (93)

On the other hand, we can easily see that

 g(X,fY)=g(fX,Y) (94)

and

 g(X,Y)=g(fX,fY)+g(wX,wY) (95)

for any X,YΓ(TM) [6].

If wX=0 for all XΓ(TM) , then M is said to be invariant sub-manifold in M˜ , i.e., F(TM(p))TM(p) for each pM . In this case, f2=I and g(fX,fY)=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 XpTM(p) at pM , we denote the slant angle between FXp and TM(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 pM and XpTM(p) [5].

Thus, invariant and anti-invariant immersions are slant immersions with slant angle θ=0 and θ=π2 , 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

 f2=λI. (96)

Furthermore, if the slant angle is θ , then it satisfies λ=cos2θ [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 Dθ and DT on M such that

(i) TM has the orthogonal direct decomposition TM=DDT.

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

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

In a semi-slant sub-manifold, if θ=π2 , 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 R7 given by the equations

 x12+x22=x52+x62, x3+x4=0. (97)

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

 z1=cosθ∂∂x1+sinθ∂∂x2+cosβ∂∂x5+sinβ∂∂x6z2=−usinθ∂∂x1+ucosθ∂∂x2, z3= ∂∂x3−∂∂x4,z4=−usinβ∂∂x5+ucosβ∂∂x6, z5= ∂∂x7, (98)

where θ,β and u denote arbitrary parameters.

For the coordinate system of R7={(x1,x2,x3,x4,x5,x6,x7)|xiR, 1i7} , we define the almost product Riemannian structure F as follows:

 F(∂∂xi)=∂∂xi, F(∂∂xj)=∂∂xj, 1≤i≤3 and 4≤j≤7. (99)

Since Fz1 and Fz3 are orthogonal to M and Fz2 ,  Fz4 ,  Fz5 are tangent to M , we can choose a D=Sp{z2,z4,z5} and D=Sp{z1,z3} . Thus, M is a 5 dimensional semi-invariant sub-manifold of R7 with usual almost Riemannian product structure (F,<,>).

Example 3.2. Let M be sub-manifold of R8 by given

 (u+v,u−v,ucosα,usinα,u+v,u−v,ucosβ,usinβ) (100)

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

 e1=∂∂x1+∂∂x2+cosα∂∂x3+sinα∂∂x4+∂∂x5−∂∂x6+cosβ∂∂x7+sinβ∂∂x8e2=∂∂x1−∂∂x2+∂∂x5+∂∂x6, e3=−usin∂∂x3+ucosα∂∂x4,e4=−usinβ∂∂x7+ucosβ∂∂x8. (101)

For the almost Riemannian product structure F of R8=R4xR4 , F(TM) is spanned by vectors

 Fe1=∂∂x1+∂∂x2+cosα∂∂x3+sinα∂∂x4−∂∂x5+∂∂x6−cosβ∂∂x7−sinβ∂∂x8,Fe2=∂∂x1−∂∂x2−∂∂x5−∂∂x6, .Fe3=e3andFe4=−e4. (102)

Since Fe1 and Fe2 are orthogonal to M and Fe3 and Fe4 are tangent to M , we can choose DT=Sp{e3,e4} and D=Sp{e1,e2} . Thus, M is a four-dimensional semi-invariant sub-manifold of R8=R4xR4 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 Dθ and D on M such that

1. The tangent bundle TM=DθD .

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

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

As a special case, if θ=0 and θ=π2 , then pseudo-slant sub-manifold becomes semi-invariant and anti-invariant sub-manifolds, respectively.

Example 3.3. Let M be a sub-manifold of R6 by the given equation

 (3u,v,vsinθ,vcosθ,scost,−scost) (103)

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

 e1=3∂∂x1, e2=∂∂y1+sinθ∂∂x2+cosθ∂∂y2,e3=cost∂∂x3−cost∂∂y3, e4=−ssint∂∂x3+ssint∂∂y3. (104)

For the almost product Riemannian structure F of R6 whose coordinate systems (x1,y1,x2,y2,x3,y3) choosing

 F(∂∂xi)=∂∂yi, 1≤i≤3,F(∂∂yj)=∂∂xj, 1≤j≤3, (105)

Then, we have

 Fe1=3∂∂y1, Fe2=−∂∂x1+sinθ∂∂y2−cosθ∂∂x2Fe3=cost∂∂y3+cost∂∂x3, Fe4=−ssint∂∂y3−ssint∂∂x3. (106)

Thus, Dθ=Sp{e1,e2} is a slant distribution with slant angle α=π4 . Since Fe3 and Fe4 are orthogonal to M , D=Sp{e3,e4} is an anti-invariant distribution, that is, M is a 4-dimensional proper pseudo-slant sub-manifold of R6 with its almost Riemannian product structure (F,<,>).

References

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