LI Kai-peng,WANG Xu-sheng
(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)
TOTALLY UMBILICAL SUBMANIFOLD ON RIEMANNIAN MANIFOLD WITH AN ORTHOGONAL CONNECTION
LI Kai-peng,WANG Xu-sheng
(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)
In this paper,we investigate the fundamental equations of submanifolds under orthogonal connections and apply the results in totally umbilical submanifolds.By using the method of Cartan to split the torsion tensor into three components,we calculate and attain the fundamental equations.We consider a special orthogonal connection with which the Riemannian curvature has the same properties as the Levi-Civita connection.We use the fundamental equations to argue totally umbilical submanifolds on spaces with constant curvature,which generalizes the results under the Levi-Civita connection.
orthogonal connections;fundamental equations in Riemannian manifolds;submanifold;umbilical point
Orthogonal connections are affine connections compatible with the metric.Cartan researched general orthogonal connections in the 1920s.An orthogonal connection minus the Levi-Civita equals a tensor which is called torsion.Cartan found that in general the torsion tensor can split into three components:the vectorial torsion,the totally anti-symmetric one and the one of Cartan-type.Taking the scalar curvature of orthogonal connections one attains the Einstein-Cartan-Hilbert functional.Its critical points are Einstein manifolds,in particular the torsion of a critical point is zero.
We review Cartan’s classi fi cation and Einstein-Cartan theory in Section 2.Under an orthogonal connection,in general,the Bianchi identity is not always hold,so many properties are not as brief as the Levi-Civita connection.We try to fi nd an orthogonal connection which is not the Levi-Civita connection satisfying the Bianchi identity.In this paper,we focus on totally umbilical submanifold in a constant curvature space.We calculate the fundamental equations,and want to use the Causs equation to express the curvature and investigate thetotally umbilical submanifold under an orthogonal connection.To read more results about orthogonal connections,especially properties on subminifolds,please refer to our other work.
We consider ann-dimensional manifoldMequipped with some Riemannian metricg.Let∇denote the Levi-Civita connection on the tangent bundle.For any affine connection∇′on the tangent bundle there exists a(2,1)-tensor fi eldAsuch that
for all vector fi eldsX,Y.
In this article we will require all connections∇′to be orthogonal,i.e.,for all vector
fieldsX,Y,Z,one has
where 〈·,·〉denotes the scalar product given by the Riemannian metricg.For any tangent vectorXone gets from(2.1)and(2.2)that the endomorphismA(X,·)is skew-adjoint
Next,we want to express some curvature quantities for∇′in terms ofAand curvature quantities for∇.To that end we fi x some pointp∈M,and we extend any tangent vectorsX,Y,Z,W∈TpMto vector fi elds again denoted byX,Y,Z,Wbeing synchronous inp,which means
Furthermore,we choose a local orthogonal frame of vector fi eldsE1,···,Enon a neighbourhood ofp,all being synchronous inp,then the Lie bracket[X,Y]=∇XY-∇YX=0 vanishes inp,and synchronicity inpimplies
Hence,inpthe Riemann tensor of∇′reads as
where Riem′denotes the Riemann tensor of∇.We note that Riem′(X,Y)Zis anti-symmetric inXandY.And by di ff erentiation of(2.3)we get that(∇EiA)(Ej,·)and(∇EjA)(Ei,·)are skewadjoint,and therefore we have
In general,Riem′does not satisfy the Bianchi identity.The Ricci curvature of∇′is defined as
by(2.4)this can be expressed as
where ric′is the Ricci curvature of∇′.We have used thatA(Ei,·)andA(X,·)are skewadjoint.
One obtains the scalar curvatureR′of∇′by taking yet another trace,inpit is given asFor the following calculation we use that(∇VA)(X,·)is skew-adjoint for any tangent vectorsV,X,and we get
whereRdenotes the scalar curvature of∇.
From(2.3)we know that the torsion tensorA(X,·)is skew-adjoint on the tangent spaceTpM.Any torsion tensorAinduces a(3,0)-tensor by setting
We define the space of all possible torsion tensors onTpMby
This vector space carries a scalar product
ForA∈Υ(TpM)andZ∈TpMone denotes the trace over the fi rst two entries by
Using the de fi nition of inner product of tensors,we denote
Theorem 2.1For dim(M)≥ 3,one has the following decomposition of(TpM)into irreducibleO(TpM)-subrepresentations
This decomposition is orthogonal with respect to 〈·,·〉,and it is given by
For dim(M)=2 theO(TpM)-representation(TpM)=1(TpM)is irreducible.
ProofStep 1Proof the decomposition exists.
Suppose anyA∈(TpM),A=A(1)+A(2)+A(3),A(i)∈i(TpM),i=1,2,3.We denoteAEiEjEkbyAijk,and denote〈Ei,Ej〉byδij,therefore
we get
soA(1)can be con fi rmed.Thenhence
ThereforeA(2)is con fi rmed.
We need to ensure thatA(3)=A-A(1)-A(2)is a Cartan-type.
For anyX,Y,Z∈TpM,since,we have
In the same way,
Add the two sides of the equations,we get,consider
Hence the decomposition exists.
Step 2The decomposition is unique.
LetA=0∈(TpM),ifA=A(1)+A(2)+A(3),then
SoV=0,i.e.,A(1)=0.
SoA(2)=0 andA(3)=0.
Step 3The three space are orthogonal with each other.
For more about this proof,cf.[12].
The connections whose torsion tensor is contained inare called vectorial.Those whose torsion tensor is in2(TpM)=∧3T∗pMare called totally anti-symmetric,and those with torsion tensor in3(TpM)are called of Cartan-type.
We note that any Cartan-type torsion tensorA∈(TpM)is trace-free in any pair of entries,i.e.,for anyZ,one has
The second equality holds asA∈(TpM),and the third one follows from the cyclic identityAXYZ+AYZX+AZXY=0.
Remark 2.2The invariant quadratic form given in(2.12)has the null space2(TpM)⊕3(TpM).More precisely,one hasA∈2(TpM)⊕3(TpM)if and only ifc12(A)(Z)=0 for anyZ∈TpM.
Remark 2.3The decomposition given in Theorem 2.1 is orthogonal with respect to the bilinear form given in(2.11),i.e.,forα,β∈{ 1,2,3},αβ,andAα∈α(TpM),Aβ∈β(TpM),one gets
Corollary 2.4For any orthogonal connection∇′on some Riemannian manifold of dimensionn≥3 there exist a vector fi eldV,a 3-formTand a(0,3)-tensor fi eldSwithSp∈3(TpM)for anyp∈Msuch thattakes the form
whereT(X,Y,·)#andS(X,Y,·)#are the unique vectors with
For any orthogonal connection theseV,T,Sare unique.
Lemma 2.5The scalar curvature of an orthogonal connection is given by
withV,T,Sas in Corollary 2.4,and div∇(V)is the divergence of the vector fi eldVtaken with respect to the Levi-Civita connection.
Corollary 2.6LetMbe a closed manifold of dimensionn≥3 with Riemannian metricgand orthogonal connection∇′.Let dvol denote the Riemannian volume measure taken with respect tog.Then the Einstein-Cartan-Hilbert functional is
LetMto be a submanifold ofThe signsare orthogonal connection,the Levi-Civita connection,torsion tensor and Riemannian curvature related toM.The signs∇′,∇,AandRare orthogonal connection,the Levi-Civita connection,torsion tensor and Riemannian curvature related toMinheriting fromand the Riemannian curvature ofM.
We have the two orthogonal decomposition
Under the Levi-Civita connection,we denoteB′byB.
It is easy to check that∇′and∇′⊥keep compatible with metric,since
And we have the fact that〈B′(X,Y),ξ〉=〈Aξ(X),Y〉.
Theorem 3.1(Guass Equation)
for anyX,Y,Z,W∈TpM.
Proof
Theorem3.2(Codazzi Equation)for anyX,Y,Z∈TpM,which
Proof
while
So the equation is found.
Theorem 3.3(Ricci Equation)
for anyX,
Proof
Proposition 3.4If(X,Y)∈TpMfor anyX,Y∈TpM,thenB′(X,Y)=B′(Y,X)=B(X,Y).
Proof
thenB′(X,Y)-B(X,Y)=0,B′(X,Y)=B(X,Y).
De fi nition 3.5We define the mean curvature vector bytrB′.If for anyX∈TM,,we callMis a submanifold with parallel mean curvature vector.
It is easy to check that ifMis a submanifold ofwith parallel mean curvature vector,we have‖H′‖is a constant.Since for any
De fi nition 3.6Mis a submanifold of,then
(1)IfAξ(x):TxM→TxMsatis fi esAξ(x)=λξ(x)·Id,whichλξ(x)is a constant related to pointx,andIdis identity mapping.Then we call x is a umbilical point related to normal vectorξ.
(2)If for allx∈M,xis a umbilical point related toξ.Then we callMis umbilical related to normal vectorξ.
(3)IfMis umbilical related to any normal vectorξ∈T⊥M.Then we callMis a totally umbilical submanifold.
Proposition 3.7LetMnto be a submanifold ofthenMis a totally umbilical submanifold if and only ifB′(X,Y)=g(X,Y)H′,∀X,Y∈TpM.
ProofIfMis a totally umbilical submanifold,then
B′(X,Y)=g(X,Y)H′.
IfB′(X,Y)=g(X,Y)H′,∀X,Y∈TpM,then∀ξ∈T⊥pM,
while〈B′(X,Y),ξ〉=〈Aξ(X),Y〉,henceAξ(X)=〈H′,ξ〉X.
Proposition 3.8LetMto be a submanifold of,thenMis a totally geodesic if and only ifMis totally umbilical andH′≡0.
ProofIfB′≡0,then
soMis totally umbilical.By Proposition 3.7,B′=gH′,thenH′≡ 0.
IfMis totally umbilical andH′≡ 0,thenB′=gH′=0,soB′≡ 0,Mis a totally geodesic submanifold.
Under the Levi-Civita connections,the Riemannian curvatureRhas the following properties
But under orthogonal connections,(iii),(iv)do not always hold.
We usually denoteG(X,Y,Z,W)〈X,Z〉〈Y,W〉-〈X,W〉〈Y,Z〉.It is easy to check thatGhas the same properties(3.1)asR.
In the rest of this section,we argue Lemma 3.9,Theorem 3.10,Corallary 3.11,and Theorem 3.12 in 3-dimensional Reimannian manifold equipped with an orthogonal connection which torsionTis a totally anti-symmetric satisfyingT=fW1∧W2∧W3(W1,W2,W3is the dual bases ofE1,E2,E3),fis a constant.
Lemma 3.9(M,g)is under the conditions above,then the fi rst Bianchi identity is founded,that is to say
ProofAt any pointp∈M,we choose parallel unit vector fi eldsE1,E2,E3as the bases in the neighborhood ofp.
Since the curvature tensorR′(X,Y)Zat pointpis not related to the extensions ofX|p,Y|p,Z|p,we let the extensions to be
whichXi,Yi,Ziare constants,i=1,2,3,then
Ifi,j,kare at least two identical,
We consideri,j,kare di ff erent from each other,then
Without of loss generality,we leti=1,j=2,k=3,
soR′(E1,E2)E3+R′(E2,E3)E1+R′(E3,E1)E2=0.That is to say,in the case ofi,j,kare di ff erent from each other,
Therefore
So property(iv)of(3.1)is founded.Since dimM=3,we have∇T=0,then
corresponding to the(iii)of(3.1).
After all,(3.1)holds for an orthogonal in 3-dimension under the conditions above.
Theorem 3.10If dimM=3,under an orthogonal connection above,then the curvature tensor ofMat pointpis determined by the all(sections’)sectional curvatures.
ProofBecuase(3.1)holds forR′,we prove the theorem as following.In order to proof the theorem,we only need to prove that if there is another(0,4)-tensor′(X,Y,Z,W)satisfying(3.1),and for any linearly independent vectorsX,Y∈TpM,it always hold that′(X,Y,X,Y)=R′(X,Y,X,Y),then for anyX,Y,Z,W∈TpM,we have′(X,Y,Z,W)=R′(X,Y,Z,W).So letS(X,Y,Z,W)=′(X,Y,Z,W)-R′(X,Y,Z,W),the argument above is equivalent to that if for anyX,Y∈TpM,S(X,Y,X,Y)=0,thenS≡0.Obviously,Sis a(0,4)-tensor satisfying(3.1).ExpandingS(X+Z,Y,X+Z,Y)=0,we have
Then expandingS(X,Y+W,Z,Y+W)=0,we have
Via(iv)S(X,Y,Z,W)+S(X,Z,W,Y)+S(X,W,Y,Z)=0,we obtain
Likewise,
Hence,for anyX,Y,Z,W∈TpM,S(X,Y,Z,W)=0.
Corollary 3.11Let(M,g)a Riemannian manifold,dimM=3,under an orthogonal connection above,thenMis a isotropic manifold if and only if fi xing anyp∈M,
Theorem 3.12is constant curvature Riemannian manifold whichdim=3 and curvature is,equipped with an orthogonal connection above,denoted by (the torsion is).LetMbe a submanifold ofwhich is connected and is totally umbilical,then
(1)Mis a submanifold with a parallel mean curvature vector,andR⊥(X,Y)ξ≡0 under the Levi-Civita connection.
(2)Mis a submanifold of constant curvature,which curvature is
ProofAt fi rst,for convenience,we proof the case ofunder the Levi-Civita connection andn-dimension
Sinceis a constant curvature manifold,then
Using Proposition 3.7,B(X,Y)=〈X,Y〉H,we have
We pickY=Z/=0,X⊥Y,thenfor anyX,Y∈TpM,,i.e.,His parallel related to
Next,proof ofR⊥≡0.
Sinceis constant curvature manifold,then.We have,whileB=B(X,λξY)=λξB(X,Y)=B(Y,Aξ(X)).
(2)Via the Gauss equation,∀X,Y,Z,W∈TpM,we have
soMis a constant curvature manifold with sectional curvature+‖H‖2,whileis a totally anti-symmetric tensor,Tis also a totally anti-symmetric tensor.Because of dim=3,dimM≤2,T≡0.So the Codazzi equation is the same as the case of the Levi-Civita connection,i.e.,
Combined with Corollary 3.11,we can get the result.
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黎曼流形在正交联络下的全脐点子流形
李凯鹏,王旭升
(武汉大学数学与统计学院,湖北武汉 430072)
本文研究了正交联络下子流形基本方程以及在全脐点子流形中的应用.利用Cartan的方法将挠率张量分解成三个部分,计算得到正交联络下的三个基本方程,并考虑一个特殊的正交联络,证明了其黎曼曲率会有类似于Levi-Civita联络下的性质.利用基本方程得到常曲率空间中的全脐点子流形的性质,推广了Levi-Civita联络下的相应结果.
正交联络;黎曼流形的基本方程;子流形;脐点
O186.12
on:53C05;53C17
A Article ID: 0255-7797(2017)04-0672-13
date:2017-01-10Accepted date:2017-03-27
Supported by National Natural Science Foundation of China(11571259).
Biography:Li Kaipeng(1989-),male,born at Fuzhou,Fujian,postguaduate,major in di ff erential geometry and geometric analysis.