Open access peer-reviewed chapter

Sub-Manifolds of a Riemannian Manifold

By Mehmet Atçeken, Ümit Yıldırım and Süleyman Dirik

Submitted: May 10th 2016Reviewed: September 23rd 2016Published: January 18th 2017

DOI: 10.5772/65948

Downloaded: 978

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 Minto an m-dimensional Riemannian manifold (M˜,g˜). Denote by g=i*g˜the induced Riemannian metric on M. Thus, ibecome an isometric immersion and Mis also a Riemannian manifold with the Riemannian metric g(X,Y)=g˜(X,Y)for any vector fields X,Yin M. The Riemannian metric gon Mis called the induced metric on M. In local components, gij=gABBjBBiAwith g=gjidxjdxjand g˜=gBAdUBdUA.

If a vector field ξpof M˜at a point pMsatisfies

g˜(Xp,ξp)=0E1

for any vector Xpof Mat p, then ξpis called a normal vector of Min M˜at p. A unit normal vector field of Min M˜is called a normal section on M[3].

By TM, we denote the vector bundle of all normal vectors of Min M˜. Then, the tangent bundle of M˜is the direct sum of the tangent bundle TMof Mand the normal bundle TMof Min M˜, i.e.,

TM˜=TMTM.E2

We note that if the sub-manifold Mis of codimension one in M˜and they are both orientiable, we can always choose a normal section ξon M, i.e.,

g(X,ξ)=0,g(ξ,ξ)=1,E3

where Xis any arbitrary vector field on M.

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

˜XY=XY+h(X,Y)E4

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

Proposition 1.1. ∇ is the Riemannian connection of the induced metric g=i*g˜on Mand h(X,Y)is a normal vector field over M, which is symmetric and bilinear in Xand Y.

Proof: Let αand βbe 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)E5

This implies that

αX(βY)=αX(β)Y+αβXYE6

and

h(αX,βY)=αβh(X,Y).E7

Eq. (6) shows that defines an affine connection on Mand Eq. (4) shows that his bilinear in Xand Ysince 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].E8

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

XYYX=[X,Y]E9

and

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

These equations show that has no torsion and his 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,Ztangent to M, that is, is also the Riemannian connection of the induced metric gon M.

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

h: Γ(TM)×Γ(TM)Γ(TM).E12

If h=0identically, then sub-manifold Mis 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 Mbe an n-dimensional sub-manifold of an m-dimensional Riemannian manifold (M˜,g˜). By h, we denote the second fundamental form of Min M˜.

H=1ntrace(h)is called the mean curvature vector of Min M˜. If H=0, the sub-manifold is called minimal.

On the other hand, Mis called pseudo-umbilical if there exists a function λon M, such that

g˜(h(X,Y),H)=λg(X,Y)E13

for any vector fields X,Yon Mand Mis 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 Mbe a sub-manifold of a Riemannian manifold (M˜,g˜)and Vbe a normal vector field on M, Xbe a vector field on M. Then, we decompose

˜XV=AVX+XV,E15

where AVXand XVdenote the tangential and the normal components of XV, respectively. We can easily see that AVXand XVare both differentiable vector fields on Mand normal bundle of M, respectively. Moreover, Eq. (15) is also called Weingarten formula.

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

(a) AVXis bilinear in vector fields Vand X. Hence, AVXat point pMdepends only on vector fields Vpand Xp.

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

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

Proof: Let αand βbe 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.E17

This implies that

AβVαX=αβAVXE18

and

αXβV=αX(β)V+αβXV.E19

Thus, AVXis bilinear in Vand X. Additivity is trivial. On the other hand, since g is a Riemannian metric,

Xg˜(Y,V)=0,E20

for any X,YΓ(TM)and VΓ(TM).

Eq. (12) implies that

g˜(˜XY,V)+g˜(Y,˜XV)=0.E21

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

g˜(h(X,Y),V)g(AVX,Y)=0.E22

The proof is completed [3].

Let Mbe a sub-manifold of a Riemannian manifold (M˜,g˜), and hand AVdenote the second fundamental form and shape operator of M,respectively.

The covariant derivative of hand AVis, respectively, defined by

(˜Xh)(Y,Z)=Xh(Y,Z)h(XY,Z)h(Y,XZ)E23

and

(XA)VY=X(AVY)AXVYAVXYE24

for any vector fields X,Ytangent to Mand any vector field Vnormal to M. If Xh=0for all X, then the second fundamental form of Mis 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).E25

Example 1.1. We consider the isometric immersion

ϕ: R2R4,E26
ϕ(x1,x2)=(x1,x121,x2,x221)E27

we note that M=ϕ(R2)R4is a two-dimensional sub-manifold of R4and 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) }.E28

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

˜e1 e1=2x1x1212x121e112x121w1,E29
˜e2 e2=2x2x2212x221e212x221w2E30

and

e2 e1=0.E31

Thus, we have h(e1,e1)=12x121w1, h(e2,e2)=12x221w2and h(e2,e1)=0.The mean curvature vector of M=ϕ(R2)is given by

H=12(w1+w2).E32

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,E33

and

g(Aw2e1,e1)=g(h(e1,e1),w2)=0,g(Aw2e1,e2)=0, g(Aw2e2,e2)=1.E34

Thus, we have

Aw1=(1000) and Aw2=(0001).E35

Now, let Mbe a sub-manifold of a Riemannian manifold (M˜,g), R˜and Rbe 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)XE36

from which

R˜(X,Y)Z=R(X,Y)Z+Ah(X,Z)YAh(Y,Z)X+(Xh)(Y,Z)(Yh)(X,Z),E37

for any vector fields X,Yand Ztangent to M. For any vector field Wtangent 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)=(Xh)(Y,Z)(Yh)(X,Z).E39

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

In particular, if M˜is of constant curvature, R˜(X,Y)Zis 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 Rof the normal bundle of the sub-manifold Mby

R(X,Y)V=XYVYXV[X,Y]VE40

for any vector fields X,Ytangent to sub-manifold M, and any vector field Vnormal 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)XE41

For any normal vector Uto 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 [AU,AV]=AUAVAVAU, Eq. (42) implies

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

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

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

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

(Xh)(Y,Z)=(Yh)(X,Z)E44

which is equivalent to

(XA)VY=(YA)VX.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)Xg(X,Z)Y}E46

for any vector fields X,Yand Zon M˜.

Since R˜(X,Y)Zis 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([AU,AV]X,Y),E48

respectively.

Proposition 1.3. A totally umbilical sub-manifold Min a real space form M˜of constant curvature cis also of constant curvature.

Proof: Since Mis 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 Mis of constant curvature c+H2for n>2. If n=2, H=constantfollows 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)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 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)}E52
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)],E53

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

Therefore, the scalar curvature rof sub-manifold Mis given by

r=cn(n1)eaTr2(Aea)eaTr(Aea)2E54

eaTr(Aea)2is 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.E55

2. Distribution on a manifold

An m-dimensional distribution on a manifold M˜is a mapping Ddefined on M˜, which assignes to each point pof M˜an m-dimensional linear subspace Dpof TM˜(p). A vector field Xon M˜belongs to Dif we have Xp Dpfor each pM˜. When this happens, we write XΓ(D). The distribution Dis 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 Dis said to be involutive if for all vector fields X,YΓ(D)we have [X,Y]Γ(D). A sub-manifold Mof M˜is said to be an integral manifold of Dif for every point p M, Dpcoincides with the tangent space to Mat p. If there exists no integral manifold of Dwhich contains M, then Mis called a maximal integral manifold or a leaf of D. The distribution Dis said to be integrable if for every pM˜, there exists an integral manifold of Dcontaining p[2].

Let ˜and distribution be a linear connection on M˜, respectively. The distribution Dis said to be parallel with respect to M˜, if we have

˜XYΓ(D)for all XΓ(TM˜) and YΓ(D)E56

Now, let (M˜,g˜)be Riemannian manifold and Dbe a distribution on M˜. We suppose M˜is endowed with two complementary distribution Dand D, i.e.,we have TM˜=DD. Denoted by Pand Qthe projections of TM˜to Dand D, respectively.

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

XY=PX'PY+QX'QY+PS(X,PY)+QS(X,QY)E57

for any X,YΓ(TM˜), where 'and Sare, 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)E58

for any X,YΓ(TM˜). We can put

X=PX+QXE59

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)E60

for any X,YΓ(TM˜).

The distributions Dand Dare both parallel with respect to if and only if we have

ϕ(XPY)=0 and P(XQY)=0. E61

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

QX'PY+QS(X,PY)=0 and PX'QY+PS(X,QY)=0. E62

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

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

FX=PXQXE63

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

(XF)Y=XFYFXYE64

for all X,YΓ(TM˜). We say that the almost product structure Fis parallel with respect to the connection , if we have XF=0. In this case, Fis called the Riemannian product structure [2].

Theorem 2.2. Let (M˜,g˜)be a Riemannian manifold and D, Dbe orthogonal distributions on M˜such that TM˜=DD.Both distributions Dand Dare parallel with respect to if and only if Fis a Riemannian product structure.

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

˜YPX= ˜PYPX+˜QYPXE65

and

˜YX=˜PYPX+˜PYQX+˜QYPX+˜QYQX, E66

from which

g(˜QYPX,QZ)=QYg(PX,QZ)g(QYQZ,PX)=0g(˜QYQZ,PX)=0,E67

that is, QYPXΓ(D)and so P˜QYPX=˜QYPX,

Q˜QYPX=0. E68

In the same way, we obtain

g(˜PYQX,PZ)=PYg(QX,PZ)g(QX,˜PYPZ)=0,E69

which implies that

P˜PYQX=0 and Q˜PYQX=˜PYQX. E70

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

P˜YX=˜PYPX+˜QYPX. E71

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

(˜ YP)X=˜ YPXP˜YX =˜PYPX+˜QYPX˜PYPX˜QYPX=0.E72

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

˜F=˜(PQ)=0.E73

This proves our assertion [2].

Theorem 2.3. Both distributions Dand Dare 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 Dand Dare parallel. Since is a torsion free linear connection, we have

[X,Y]=XYYXΓ(D),for anyX,YΓ(D)E74

and

[U,V]=UVVUΓ(D),for anyU,VΓ(D)E75

Thus, Dand Dare integrable distributions. Now, let Mbe a leaf of Dand denote by hthe second fundamental form of the immersion of Min M˜. Then by the Gauss formula, we have

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

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

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

XYΓ(D) for any X,YΓ(D)E77

and

UVΓ(D) for any U,VΓ(D).E78

Since gis a Riemannian metric tensor, we obtain

g(UY,V)=g(Y,UV)=0E79

and

g(XV,Y)=g(V,XY)=0E80

for any X,YΓ(D)and U,VΓ(D).Thus, both distributions Dand Dare parallel on M˜.

3. Locally decomposable Riemannian manifolds

Let (M˜,g˜)be ndimensional Riemannian manifold and Fbe 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)E81

for any X,YΓ(TM˜)then M˜is called almost Riemannian product manifold and Fis said to be almost Riemannian product structure. If Fis 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).E82

Then, we have

P+Q=I, P2=P, Q2=Q, PQ=QP=0 and F=PQ.E83

Thus, Pand Qdefine two complementary distributions Pand Qglobally. Since F2=I, we easily see that the eigenvalues of Fare 1and 1. An eigenvector corresponding to the eigenvalue 1is in Pand an eigenvector corresponding to 1is in Q. If Fhas eigenvalue 1of multiplicity Pand eigenvalue 1of multiplicity q, then the dimension of Pis pand that of Qis q. Conversely, if there exist in M˜two globally complementary distributions Pand Qof dimension pand q, respectively. Then, we can define an almost Riemannian product structure Fon 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 Pand Qby Mpand Mq, respectively. Then we can write M˜=MpXMq, (p,q>2). Also, we denote the components of the Riemannian curvature Rof M˜by Rdcba, 1a,b,c,dn=p+q.

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

Rdcba=λ{gdagcbgcagdb}E84

and

Rzyxw=μ{gzwgyxgywgzx}.E85

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

Rkjih=14(λ+μ){(gkhgjigjhgki)+(FkhFjiFjhFki)}+14(λμ){(FkhgjiFjhgki)+(gkhFjigjhFki)}.E86

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)}.E87

Then, we have

Rcdba=2(a+b){gdagcbgcagdb}E88

and

Rzyxw=2(ab){gzwgyxgywgzx}.E89

Let M˜be an mdimensional almost Riemannian product manifold with the Riemannian structure (F,g˜)and Mbe an ndimensional sub-manifold of M˜. For any vector field Xtangent to M, we put

FX=fX+wX,E90

where fXand wXdenote 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,E91

where BVand CVdenote the tangential and normal components of FV, respectively.

Then, we have

f2+Bw=I,Cw+wf=0 E92

and

fB+BC=0, wB+C2=I.E93

On the other hand, we can easily see that

g(X,fY)=g(fX,Y)E94

and

g(X,Y)=g(fX,fY)+g(wX,wY)E95

for any X,YΓ(TM)[6].

If wX=0for all XΓ(TM), then Mis said to be invariant sub-manifold in M˜, i.e., F(TM(p))TM(p)for each pM. In this case, f2=Iand 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=0and hence Mis an invariant sub-manifold in M˜.

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

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

Definition 3.1. Let Mbe 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 FXpand TM(p)by θ(p). Then Msaid to be slant sub-manifold if the angle θ(p)is constant, i.e.,it is independent of the choice of pMand XpTM(p)[5].

Thus, invariant and anti-invariant immersions are slant immersions with slant angle θ=0and θ=π2, respectively. A proper slant immersion is neither invariant nor anti-invariant.

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

f2=λI.E96

Furthermore, if the slant angle is θ, then it satisfies λ=cos2θ[9].

Definition 3.2. Let Mbe a sub-manifold of an almost Riemannian product manifold M˜with almost Riemannian product structure (F,g˜). Mis said to be semi-slant sub-manifold if there exist distributions Dθand DTon Msuch that

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

(ii) The distribution Dθis a slant distribution with slant angle θ.

(iii) The distribution DTis 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 Min R7given by the equations

x12+x22=x52+x62, x3+x4=0.E97

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

z1=cosθx1+sinθx2+cosβx5+sinβx6z2=usinθx1+ucosθx2, z3= x3x4,z4=usinβx5+ucosβx6, z5= x7,E98

where θ,βand udenote arbitrary parameters.

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

F(xi)=xi, F(xj)=xj, 1i3 and 4j7.E99

Since Fz1and Fz3are orthogonal to Mand Fz2, Fz4, Fz5are tangent to M, we can choose a D=Sp{z2,z4,z5}and D=Sp{z1,z3}. Thus, Mis a 5dimensional semi-invariant sub-manifold of R7with usual almost Riemannian product structure (F,<,>).

Example 3.2. Let Mbe sub-manifold of R8by given

(u+v,uv,ucosα,usinα,u+v,uv,ucosβ,usinβ)E100

where u,vand βare the arbitrary parameters. By direct calculations, we can easily see that the tangent bundle of Mis spanned by

e1=x1+x2+cosαx3+sinαx4+x5x6+cosβx7+sinβx8e2=x1x2+x5+x6, e3=usinx3+ucosαx4,e4=usinβx7+ucosβx8.E101

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

Fe1=x1+x2+cosαx3+sinαx4x5+x6cosβx7sinβx8,Fe2=x1x2x5x6, .Fe3=e3andFe4=e4.E102

Since Fe1and Fe2are orthogonal to Mand Fe3and Fe4are tangent to M, we can choose DT=Sp{e3,e4}and D=Sp{e1,e2}. Thus, Mis a four-dimensional semi-invariant sub-manifold of R8=R4xR4with usual Riemannian product structure F.

Definition 3.3. Let Mbe a sub-manifold of an almost Riemannian product manifold M˜with almost Riemannian product structure (F,g˜). Mis said to be pseudo-slant sub-manifold if there exist distributions Dθand Don Msuch that

  1. The tangent bundle TM=DθD.

  2. The distribution Dθis a slant distribution with slant angle θ.

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

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

Example 3.3. Let Mbe a sub-manifold of R6by the given equation

(3u,v,vsinθ,vcosθ,scost,scost)E103

where u,v,sand tarbitrary parameters and θis a constant.

We can check that the tangent bundle of Mis spanned by the tangent vectors

e1=3x1, e2=y1+sinθx2+cosθy2,e3=costx3costy3, e4=ssintx3+ssinty3.E104

For the almost product Riemannian structure Fof R6whose coordinate systems (x1,y1,x2,y2,x3,y3)choosing

F(xi)=yi, 1i3,F(yj)=xj, 1j3,E105

Then, we have

Fe1=3y1, Fe2=x1+sinθy2cosθx2Fe3=costy3+costx3, Fe4=ssinty3ssintx3.E106

Thus, Dθ=Sp{e1,e2}is a slant distribution with slant angle α=π4. Since Fe3and Fe4are orthogonal to M, D=Sp{e3,e4}is an anti-invariant distribution, that is, Mis a 4-dimensional proper pseudo-slant sub-manifold of R6with its almost Riemannian product structure (F,<,>).

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Mehmet Atçeken, Ümit Yıldırım and Süleyman Dirik (January 18th 2017). Sub-Manifolds of a Riemannian Manifold, Manifolds - Current Research Areas, Paul Bracken, IntechOpen, DOI: 10.5772/65948. Available from:

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