# A Basic Inequality for the Tanaka-Webster Connection

Authors: Dae Ho Jin; Jae Won Lee
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

For submanifolds tangent to the structure vector field in Sasakian
space forms, we establish a Chen's basic inequality between the main intrinsic invariants of the submanifold (namely, its pseudosectional curvature
and pseudosectional curvature on one side) and the main extrinsic invariant
(namely, squared pseudomean curvature on the other side) with respect to
the Tanaka-Webster connection. Moreover, involving the pseudo-Ricci curvature and the squared pseudo-mean curvature, we obtain a basic inequality
for submanifolds of a Sasakian space form tangent to the structure vector
field in terms of the Tanaka-Webster connection.


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## Body

## 1. Introduction

One of the basic interests in the submanifold theory is to establish simple relationship between intrinsic invariants and extrinsic invariants of a submanifold. Gauss-Bonnet Theorem, Isoperimetric inequality, and Chern-Lashof Theorem are those such kind of study.Chen [1] established a nice basic inequality-related intrinsic quantities and extrinsic ones of submanifolds in a space form with arbitrary codimension. Moreover, he studied the basic inequalities of submanifolds of complex space forms and characterize submanifolds when the equality holds.In this paper, we introduce pseudosectional curvatures and pseudo-Ricci curvature for the Tanaka-Webster connection in a Sasakian space form. After then, we study basic inequalities for submanifolds of a Sasakian space form of a constant pseudosectional curvature and a pseudo-Ricci curvature in terms of the Tanaka-Webster connection.


## 2. Preliminaries

LetM̃ be an odd-dimensional Riemannian manifold with a Riemannian metric g̃ satisfying
(2.1)η(ξ)=1,φ2=-I+η⊗ξ,η(X)=g̃(X,ξ),g̃(φX,φY)=g̃(X,Y)-η(X)η(Y).Then(φ,ξ,η,g̃) is called the almost contact metric structure on M̃. Let Φ denote the fundamental 2 form in M̃ given by Φ(X,Y)=g̃(X,φY) for all X,Y∈TM̃, the set of vector fields of M̃. If Φ=dη, then M̃ is said to be a contact metric manifold. Moreover, if ξ is a Killing vector field with respect to g̃, and the contact metric structure is called a 𝒦-contact structure. Recall that a contact metric manifold is 𝒦-contact if and only if (2.2)∇̃Xξ=-φX
for any X∈TM̃, where ∇̃ is the Levi-Civita connection of M̃. The structure of M̃ is said to be normal if [φ,φ]+2dη⊗ξ-0, where [φ,φ] is the Nijenhuis torsion of φ. A Sasakian manifold is a normal contact metric manifold. In fact, an almost contact metric structure is Sasakian if and only if (2.3)(∇̃Xφ)Y=g̅(X,Y)ξ-η(Y)X
for all vector fields X and Y. Every Sasakian manifold is a 𝒦-contact manifold.Given a Sasakian manifoldM̃, a plane section π in TpM̃ is called a φ-section if it is spanned by X and φX, where X is a unit tangent vector field orthogonal to ξ. The sectional curvature K̃(π) of a φ-section π is called φ-sectional curvature. If a Sasakian manifold M̃ has constant φ-sectional curvature c,M̃ is called a Sasakian space form, denoted by M̃(c). (For more details, see [2]).Now letM be a submanifold immersed in (M̃,φ,ξ,η,g). We also denote by g the induced metric on M. Let TM be the Lie algebra of vector fields in M and T⊥M the set of all vector fields normal to M. We denote by h the second fundamental form of M and by Av the Weingarten endomorphism associated with any v∈T⊥M. We put hijr=g̅(h(ei,ej),er) for any orthonormal vector ei, ej∈TM and er∈T⊥M. The mean curvature vector field H is defined by H=(1/dimM)trace(h). M is said to be totally geodesic if the second fundamental form vanishes identically.From now on, we assume that the dimension ofM is n+1, and that of the ambient manifold M̃ is 2m+1(m≥2). We also assume that the structure vector field ξ is tangent to M. Hence, if we denote by D the orthogonal distribution to ξ in TM, we have the orthogonal direct decomposition of TM by TM=D⊕span{ξ}. For any X∈TM, we write φX=TX+NX, where TX (NX, resp.) is the tangential (normal, resp.) component of φX. If φM is a 𝒦-contact manifold, (2.2) gives
(2.4)h(X,ξ)=-NX,
for any X in TM. Given a local orthonormal frame {e1,…,en} of D, we can define the squared norms of T and N by (2.5)‖T‖2=∑i,j=1ng(ei,Tej)2,‖N‖2=∑i,j=1ng(ei,Nej)2,
resepectively. It is easy to show that both ∥T∥2 and ∥N∥2 are independent of the choice of the orthonormal frames. The submanifold M is said to be invariant if N is identically zero, that is, φX∈TM for any X∈TM. On the other hand, M is said to be an anti-invariant submanifold if T is identically zero, that is, φX∈T⊥M for any X∈TM.


## 3. The Tanaka-Webster Connection for Sasakian Space Form

The Tanaka-Webster connection [3, 4] is the canonical affine connection defined on a nondegenerate pseudo-Hermitian CR-manifold. Tanno [5] defined the Tanaka-Webster connection for contact metric manifolds by the canonical connection which coincides with the Tanaka-Webester connection if the associated CR-structure is integrable. We define the Tanaka-Webster connection for submanifolds of Sasakian manifolds by the naturally extended affine connection of Tanno's Tanaka-Webster connection. Now we recall the Tanaka-Webster connection ∇̂ for contact metric manifolds
(3.1)∇̂XY=∇̃XY+η(X)φY+(∇Xη)(Y)ξ-η(Y)∇Xξ,
for all vector fields X,Y∈TM̃. Together with (2.1), ∇̂ is written by (3.2)∇̂XY=∇̃XY+η(X)φY+η(Y)φX-g̃(Y,φ(X))ξ.
Also, by using (2.1) and (2.3), we can see that (3.3)∇̂η=0,∇̂ξ=0,∇̂φ=0,∇̂g̃=0.We define the Tanaka-Webster curvature tensor ofR̃ (in terms of ∇̃) by (3.4)R̂(X,Y)Z=∇̂X∇̂YZ-∇̂Y∇̂YZ-∇̂[X,Y]Z,
for all vector fields X,Y, and Z in M̃.LetM̃(c) be a Sasakian space form of constant sectional curvature c and M a submanifold of M̃(c). Then, we have the following Gauss’ equation: (3.5)R̂(X,Y)Z=c+34[{g(Y,Z)-η(Y)η(Z)}X-{g(X,Z)-η(X)η(Z)}Y+{g(X,Z)η(Y)-g(Y,Z)η(X)}ξ+2g(X,φY)φZ]+c+74{g(Z,φY)φX-g(Z,φX)φY}
for any tangent vector fields X,Y,Z tangent to M.Let us define the connection∇° on M induced from the Tanaka-Webster connection ∇̂ on M̃ given by
(3.6)∇̅°XY=∇°XY+ĥ(X,Y),
for any X,Y∈Γ(TM), where ĥ is called the lightlike second fundamental form of M with respect to the induced connection ∇°. In the view of (3.2) and (3.6), (3.7)∇°XY+ĥ(X,Y)=∇XY+h(X,Y)+η(X)φY+η(Y)φX-g̅(Y,φX)ξ.
From (3.7), we obtain (3.8)∇°XY=∇XY+η(X)TY+η(Y)TX-g̅(Y,φX)ξ,(3.9)ĥ(X,Y)=h(X,Y)+η(X)NY+η(Y)NX,
where φX=TX+NX.From (3.3), (3.8), and (3.9) it is easy to verify the following:
(3.10)∇°η=0,∇°ξ=0,∇°φ=0,∇°g=0.
Moreover, for the induced connection ∇, we have the following (3.11)∇Xξ=-TX,h(X,ξ)=-NX.
From the definition of R̂, together with (3.5), we have (3.12)g(R°(X,Y)Z,W)=c+34[{g(Y,Z)-η(Y)η(Z)}g(X,W)-{g(X,Z)-η(X)η(Z)}g(Y,W)+{g(X,Z)η(Y)-g(Y,Z)η(X)}g(ξ,W)+2g(X,φY)g(φZ,W)]+c+74{g(Z,φY)g(φX,W)-g(Z,φX)g(φY,W)}+g̅(ĥ(X,W),ĥ(Y,Z))-g̅(ĥ(X,Z),ĥ(Y,W)),
for any X,Y,Z,W∈TM.For an orthonormal basis{e1,…,en+1} of the tangent space TpM,p∈M, the pseudoscalar curvature τ̂ at p is defined by (3.13)τ̂=∑i<jK̂(ei∧ej),
where K̂(ei∧ej) denotes the pseudosectional curvature of M associated with the plane section spanned by ei and ej for the Tanaka-Webster connection ∇̂. In particular, if we put en+1=ξp, then (3.13) implies that (3.14)2τ̂=∑i≠jK̂(ei∧ej)+2∑i=1nK̂(ei∧ξ).
Moreover, from (3.9), we have (3.15)ĥijr=hij,i,j∈{1,…,n},ĥin+1r=0,j∈{1,…,n+1}.
The pseudomean curvature vector field H is defined by Ĥ=(1/dimM)trace(ĥ). M is said to be totally pseudogeodesic if the second fundamental ĥ form vanishes identically. From (2.5), (3.12) and (3.14), we obtain the following relationship between the pseudoscalar curvature and the pseudomean curvature of M, (3.16)2τ̂=(n+1)2‖Ĥ‖2-‖ĥ‖2+n(n-1)c+34+3c+134‖T‖2.We now recall the Chen's lemma.Lemma 3.1 (see [6]).
Leta1,…,an,c be n+1(n≥2) real numbers such that
(3.17)(∑i=1nai)2=(n-1)(∑i=1nai2+c).
Then, 2a1a2≥c, with the equality holding if and only if a1+a2=a3=⋯=an.Letp∈M and let π be a plane section of TpM which is generated by orthonormal vectors X and Y. We can define a function α(π) of tangent space TpM into [0,1] by (3.18)α(π)=g(TX,Y)2,
which is well defined.Now, we prove the following.Theorem 3.2.
LetM be an  (n+1)-dimensional  (n≥2) submanifold isometrically immersed in a  m-dimensional Sasakian space form  M̃(c) such that the structure vector field  ξ is tangent to  M in terms of the Tanaka-Wester connection  ∇̂. Then, for each point  p∈M and each plane section  π⊂TpM, we have the following:
(3.19)τ̂-K̂(π)≤(n+1)2(n-1)2n‖Ĥ‖2+18(n+1)(n+2)(c+3)+3c+138‖T‖2-3c+134α(π).
Equality in (3.19) holds at p∈M if and only if there exist an orthonormal basis {e1,…,en+1} of TpM and an orthonormal basis {en+2,…,em} of Tp⊥M such that (a)  π=Span{e1,e2} and (b) the shape operators Ar=Aer, r=n+2,…,m, take the following forms:
(3.20)A=(a000-a0000n-1),A=(h11rh12r0h12r-h11r0000n-1),r=n+3,…,m.Proof.
LetMn+1 be a submanifold of M̃(c). We introduce
(3.21)ρ̂=2τ̂-(n+1)2(n-1)2n‖Ĥ‖2-(n+1)(n+2)c+34-3c+134‖T‖2.
Then, from (3.16) and (3.21), we get
(3.22)(n+1)‖Ĥ‖2=n‖ĥ‖2+n(ρ̂-2(c+3)4).
Let p be a point of M and let π⊂TpM be a plane section at p. We choose an orthonormal basis {e1,…,en+1} for TpM and {en+2,…,em} for Tp⊥M such that en+1=ξ, π= Span{e1,e2}, and the pseudomean curvature vector Ĥ is parallel to en+2. Then, from (3.22), we get
(3.23)(∑i=1n+1ĥiin+2)2=n(∑i=1n+1(ĥiin+2)2+∑i≠j(ĥijn+2)2+∑r=n+3m∑i,j(ĥijr)2+ρ̂-2(c+3)4)
and so, by applying Lemma 3.1, we obtain
(3.24)2ĥ11n+2ĥ22n+2≥∑i≠j(ĥijn+2)2+∑r=n+3m∑i,j(ĥijr)2+ρ̂-2(c+3)4.
On the other hand, from (3.12), we have
(3.25)K̂(π)=ĥ11n+2ĥ22n+2-(ĥ12n+2)2+∑r=n+3m(ĥ11rĥ22r-(ĥ12r)2)c+34+3c+134g2(e1,φe2).
Then, from (3.24) and (3.25), we get
(3.26)K̂(π)=ρ2+3c+134g2(e1,φe2)+∑r=n+2m∑j>2((ĥ1jr)2+(ĥ2jr)2)+12∑i≠j>2(ĥijn+2)2(3.27)+12∑r=n+3m∑i,j>2(ĥijr)2+12∑r=n+3m(ĥ11r+ĥ22r)2≥ρ2+3c+134α(π).
Combining (3.21) and (3.27), the inequality (3.19) yields. If the equality in (3.19) holds, then the inequalities given by (3.24) and (3.27) become equalities. In this case, we have
(3.28)ĥ1jn+2=ĥ2jn+2=ĥijn+2=0,i≠j>2,ĥ1jr=ĥ2jr=ĥijr=0,r∈{n+3,…,m};i,j∈{3,…,n+1},ĥ11n+3+ĥ22n+3=⋯=ĥ11m+ĥ22m=0.
Moreover, choosing e1 and e2 such that h12n+2=0, from (3.11), we also have the following
(3.29)ĥ11n+2+ĥ22n+2=ĥ33n+2=⋯=ĥn+1n+1n+2=0.
Thus, with respect to the chosen orthonormal basis {e1,…,em}, the shape operators of M take the forms.We now define a well-defined functionδ̂M on M by using (infK̂)(p)=inf{K̂(π)∣πisaplanesection⊂TpM} in the following manner: (3.30)δ̂M=τ̂-inf⁡⁡K̂.
If c=-13/3, then we obtain directly from (3.19) the following result.Corollary 3.3.
LetM be an (n+1)-dimensional (n≥2) submanifold isometrically immersed in a m-dimensional Sasakian space form M̃(-13/3) such that the structure vector field ξ is tangent to M in terms of the Tanaka-Wester connection ∇̂. Then, for each point p∈M and each plane section π⊂TpM, we have the following:
(3.31)δ̂M≤(n+1)2(n-1)2n‖Ĥ‖2-16(n+1)(n+2).
The equality in (3.31) holds if and only if M is a anti-invariant submanifold with rank(T)=2.Proof.
In order to estimateδ̂M, we minimize ∥T∥2-2α(π) in (3.19). For an orthonormal basis {e1,…,en+1} of TpM with π=Span{e1,e2}, we write
(3.32)‖T‖2-2α(π)=∑i,j=3n+1g2(ei,φej)+2∑j=3n+1{g2(e1,φej)+g2(e2,φej)}.
Thus, we see that the minimum value of ∥T∥2-2α(π) is zero, provided that π=Span{e1,e2} is orthogonal to ξ, and span{φej∣j=3,…,n} is orthogonal to TpM. Thus we have (3.31) with equality case holding if and only if M is anti-invariant such that rank(T)=2.


## 4. A Pseudo-Ricci Curvature for Sasakian Space Form

We denote the set of unit vectors inTpM by Tp1M by (4.1)Tp1M={X∈TpM∣g(X,X)=1}.
Let {e1,…,ek}, 2≤k≤n, be an orthonormal basis of a k-place section Πk of TpM. If k=n, then Πk=TpM, and if k=2, then Π2 is a plane section of TpM. For a fixed i∈{1,…,k}, a k-pseudo-Ricci curvature of Πk at ei, denoted by RiĉΠk(ei), is defined by [7] (4.2)RiĉΠk(ei)=∑j≠ikK̂ij,
where K̂ij is the pseudosectional curvature in terms of the Tanaka-Webster connection ∇̂ of the plane section spanned by ei and ej. We note that an n-pseudo-Ricci curvature RicTpM(ei) is the usual pseudo-Ricci curvature of ei, denoted by Riĉ(ei). Thus, for any orthonormal basis {e1,…,en+1} for TpM and for a fixed i∈{1,…,n+1}, we have the following: (4.3)RiĉTpM(ei)=Riĉ(ei)=∑j≠in+1K̂ij.
The pseudoscalar curvature τ̂(Πk) of the k-plane section Πk is given by (4.4)τ̂(Πk)=∑1≤i<j≤n+1K̂ij.
The relative null spae of M at p is defined by [8] (4.5)Np={X∈TpM∣ĥ(X,Y)=0,∀Y∈TpM}.Theorem 4.1.
LetM̃(c) be a m-dimensional Sasakian space form and M an n+1-dimensional submanifold tangent to ξ with respect to the Tanaka-Webster connection ∇̂. Then,  (i)
for each unit vectorX∈TpM orthogonal to ξ, we have
(4.6)4Riĉ(X)≤(n+1)2‖Ĥ‖2+(n-1)(c+3)+(3c+13)‖TX‖2,(ii)
ifĤ(p)=0, then a unit tanget vector X∈TpM orthogonal to ξ satisfies the equality case of (4.6) if and only of X∈𝒩p.(iii)
the equality case of (4.6) holds identically for all unit tangent vectors orthogonal to ξ at p if and only if p is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.Proof.
(i) LetX∈TpM be a unit tangent vector at p, orthogonal to ξ. We choose an orthonormal basis {e1,…,en+1} for TpM and {en+2,…,em} for Tp⊥M such that e1=X and en+1=ξ. Then, from (3.16), we have
(4.7)(n+1)2‖Ĥ‖2=2τ̂+‖ĥ‖2-n(n-1)c+34-3c+134‖T‖2.
From (4.7), we get
(4.8)(n+1)2‖Ĥ‖2=2τ̂+∑r=n+2m[(ĥ112)2+(ĥ22r+⋯+ĥn+1n+1r)2+2∑i<j(ĥijr)2]-2∑r=n+2m∑2≤i<j≤nĥiirĥjjr-n(n-1)c+34-3c+134‖T‖2=2τ̂+12∑r=n+2m[(ĥ11r+ĥ22r+⋯+ĥn+1n+1r)2+(ĥ11r-ĥ22r-⋯-ĥn+1n+1r)2]+2∑n+2m∑i<j(ĥijr)2-2∑r=n+2m∑2≤i<j≤nĥiirĥjjr-n(n-1)c+34-3c+134‖T‖2.
From (3.12), we have
(4.9)K̂ij=∑r=n+2m[ĥiirĥjjr-(ĥijr)2]+c+34+3c+134g2(ei,Tej),
and consequently
(4.10)∑2≤i<j≤n+1K̂ij=∑r=n+2m[ĥiirĥjjr-(ĥijr)2]+(n-1)(n-2)(c+3)8+3c+138{‖T‖2-2‖Te1‖2}.
Substituting (4.10) into (4.8), one gets
(4.11)(n+1)2‖Ĥ‖2≥2τ̂+(n+1)22‖Ĥ‖2+2∑r=n+2m∑j=2(ĥ1jr)2-2∑2≤i<j≤n+1K̂ij-(n-1)(c+3)2-3c+132‖Te1‖2.
Therefore,
(4.12)(n+1)22‖Ĥ‖2≥2Riĉ(X)-(n-1)(c+3)2-3c+132‖TX‖2,
which is equivalent to (4.6)
(ii) Assume thatĤ(p)=0. Equality holds in (4.6) if and only if
(4.13)ĥ12r=⋯=ĥ1n+1r=0,ĥ11r=ĥ22r+⋯+ĥn+1n+1r,r∈{n+2,…,m}.
Then, ĥ1jr=0 for each j∈{1,…,n+1},r∈{n+2,…,m}, that is, X∈𝒩p.
(iii) The equality case of (4.6) holds for all unit tangent vectors at p if and only if
(4.14)ĥijr=0,i≠j,r∈{n+2,…,m},ĥ11r+⋯+ĥn+1n+1r-2ĥiir=0,i∈{1,…,n+1},r∈{n+2,…,m}.
Since ĥ(ei,en+1=ξ)=0 from (3.10), p is a totally pseudogeodesic point, and, hence, φ(TpM)⊂TpM. The converse is trivial.Corollary 4.2.
LetM be an n+1-dimensional invariant submanifold of a Sasakian space form M̃(c). Then,  (i)
for each unit vectorX∈TpM orthogonal to ξ, we have
(4.15)4Riĉ(X)≤(n-1)(c+3)+(3c+13).(ii)
A unit tanget vectorX∈TpM orthogonal to ξ satisfies the equality case of (4.6) if and only if X∈𝒩p.(iii)
The equality case of (4.6) holds identically for all unit tangent vectors orthogonal to ξ at p if and only if p is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.



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