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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).

Chapter 3

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

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˜=TMTM.
(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+βXY+βh(X,Y)} αXβY+h(αX,βY)=αβXY+αX(β)Y+αβh(X,Y)
(5)

This implies that

αX(βY)=αX(β)Y+αβXY
(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)XYh(Y,X)[X,Y].
(8)

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

XYYX=[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(TM).
(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+XV,
(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+αXβV=αX(β)VαβAVX+αβXV.
(17)

This implies that

AβVαX=αβAVX
(18)

and

αXβV=αX(β)V+αβXV.
(19)

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

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)=Xh(Y,Z)h(XY,Z)h(Y,XZ)
(23)

and

(XA)VY=X(AVY)AXVYAVXY
(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: R2R4,
(26)
ϕ(x1,x2)=(x1,x121,x2,x221)
(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

TM=sp{w1=(x1,x121,0,0),w2=(0,0,x1,x221) }.
(28)

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

˜e1 e1=2x1x1212x121e112x121w1,
(29)
˜e2 e2=2x2x2212x221e212x221w2
(30)

and

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)=12x121(x12+x121)=1,g(Aw1e2,e2)=g(h(e2,e2),w1)=12x221g(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=(0001).
(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]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
(36)

from which

R˜(X,Y)Z=R(X,Y)Z+Ah(X,Z)YAh(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=XYVYXV[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]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
(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)Xg(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(n1)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(n1)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

˜XYΓ(D)for all XΓ(TM˜) and YΓ(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=PX'PY+QX'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

for any XΓ(TM˜) . 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)
(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

QX'PY+QS(X,PY)=0 and PX'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

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=XFYFXY
(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)=0g(˜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=˜ YPXP˜YX =˜PYPX+˜QYPX˜PYPX˜QYPX=0.
(72)

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

˜F=˜(PQ)=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]=XYYXΓ(D),for anyX,YΓ(D)
(74)

and

[U,V]=UVVUΓ(D),for anyU,VΓ(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

XYΓ(D) for any X,YΓ(D)
(77)

and

UVΓ(D) for any U,VΓ(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(IF).
(82)

Then, we have

P+Q=I, P2=P, Q2=Q, PQ=QP=0 and F=PQ.
(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{gdagcbgcagdb}
(84)

and

Rzyxw=μ{gzwgyxgywgzx}.
(85)

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

Rkjih=14(λ+μ){(gkhgjigjhgki)+(FkhFjiFjhFki)}+14(λμ){(FkhgjiFjhgki)+(gkhFjigjhFki)}.
(86)

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

Rkjih=14(λ+μ){(gkhgjigjhgki)+(FkhFjiFjhFki)}+14(λμ){(FkhgjiFjhgki)+(gkhFjigjhFki)}.
(87)

Then, we have

Rcdba=2(a+b){gdagcbgcagdb}
(88)

and

Rzyxw=2(ab){gzwgyxgywgzx}.
(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

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

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

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=cosθx1+sinθx2+cosβx5+sinβx6z2=usinθx1+ucosθx2, z3= x3x4,z4=usinβx5+ucosβx6, 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, 1i3 and 4j7.
(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,uv,ucosα,usinα,u+v,uv,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+cosαx3+sinαx4+x5x6+cosβx7+sinβx8e2=x1x2+x5+x6, e3=usinx3+ucosαx4,e4=usinβx7+ucosβx8.
(101)

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

Fe1=x1+x2+cosαx3+sinαx4x5+x6cosβx7sinβx8,Fe2=x1x2x5x6, .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=3x1, e2=y1+sinθx2+cosθy2,e3=costx3costy3, e4=ssintx3+ssinty3.
(104)

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

F(xi)=yi, 1i3,F(yj)=xj, 1j3,
(105)

Then, we have

Fe1=3y1, Fe2=x1+sinθy2cosθx2Fe3=costy3+costx3, Fe4=ssinty3ssintx3.
(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

1 - Katsuei Kenmotsu, editor. Differential Geometry of Submaifolds. Berlin: Springe-Verlag; 1984. 134 p.
2 - Aurel Bejancu. Geometry of CR-Submanifolds. Dordrecht: D. Reidel Publishing Company; 1986. 172 p. DOI: QA649.B44
3 - Bang-Yen Chen. Geometry of Submanifolds. New York: Marcel Dekker, Inc.; 1973. 298 p.
4 - Kentaro Yano and Masahiro Kon. Structurs on Manifolds. Singapore: World Scientific Publishing Co. Pte. Ltd.; 1984. 508 p. DOI: QA649.Y327
5 - Meraj Ali Khan. Geometry of Bi-slant submanifolds “Some geometric aspects on sub-manifolds Theory”. Saarbrücken, Germany: Lambert Academic Publishing; 2006. 112 p.
6 - Mehmet Atçeken. Warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. Mathematical Problems in Engineering. 2009;2009:621625. DOI: doi:10.1155/2009/621625
7 - Tyuzi Adati. Submanifolds of an almost product Riemannian manifold. Kodai Mathematical Journal. 1981;4(2):327–343.
8 - Mehmet Atçeken. A condition for warped product semi-invariant submanifolds to be Riemannian product semi-invariant Sub-manifoldsub-manifolds in locally Riemannian product manifolds. Turkish Journal of Mathematics. 2008;33:349–362.
9 - Mehmet Atçeken. Slant submanifolds of a Riemannian product manifold. Acta Mathematica Scientia. 2010;30(1):215–224. DOI: doi:10.1016/S0252-9602(10)60039-2