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.

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

Let M̃ be an odd-dimensional Riemannian manifold with a Riemannian metric g̃ satisfying
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 ∇̃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 (∇̃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 manifold M̃, 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 let M 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 of M 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
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 ‖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
∇̂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 ∇̂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 ∇̂η=0,∇̂ξ=0,∇̂φ=0,∇̂g̃=0.

We define the Tanaka-Webster curvature tensor of R̃ (in terms of ∇̃) by R̂(X,Y)Z=∇̂X∇̂YZ-∇̂Y∇̂YZ-∇̂[X,Y]Z,
for all vector fields X,Y, and Z in M̃.

Let M̃(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: 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
∇̅°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), ∇°XY+ĥ(X,Y)=∇XY+h(X,Y)+η(X)φY+η(Y)φX-g̅(Y,φX)ξ.
From (3.7), we obtain ∇°XY=∇XY+η(X)TY+η(Y)TX-g̅(Y,φX)ξ,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:
∇°η=0,∇°ξ=0,∇°φ=0,∇°g=0.
Moreover, for the induced connection ∇, we have the following ∇Xξ=-TX,h(X,ξ)=-NX.
From the definition of R̂, together with (3.5), we have 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 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 2τ̂=∑i≠jK̂(ei∧ej)+2∑i=1nK̂(ei∧ξ).
Moreover, from (3.9), we have 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, 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 [<xref ref-type="bibr" rid="B3">6</xref>]).

Let a1,…,an,c be n+1(n≥2) real numbers such that
(∑i=1nai)2=(n-1)(∑i=1nai2+c).
Then, 2a1a2≥c, with the equality holding if and only if a1+a2=a3=⋯=an.

Let p∈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 α(π)=g(TX,Y)2,
which is well defined.

Now, we prove the following.

Theorem 3.2.

Let M 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:
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:
A=(a000-a0000n-1),A=(h11rh12r0h12r-h11r0000n-1),r=n+3,…,m.

Proof.

Let Mn+1 be a submanifold of M̃(c). We introduce
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
(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
(∑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
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
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
K̂(π)=ρ2+3c+134g2(e1,φe2)+∑r=n+2m∑j>2((ĥ1jr)2+(ĥ2jr)2)+12∑i≠j>2(ĥijn+2)2+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
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
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 function δ̂M on M by using (infK̂)(p)=inf{K̂(π)∣πisaplanesection⊂TpM} in the following manner: δ̂M=τ̂-infK̂.
If c=-13/3, then we obtain directly from (3.19) the following result.

Corollary 3.3.

Let M 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:
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 estimate δ̂M, 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
‖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 in TpM by Tp1M by 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] 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: RiĉTpM(ei)=Riĉ(ei)=∑j≠in+1K̂ij.
The pseudoscalar curvature τ̂(Πk) of the k-plane section Πk is given by τ̂(Πk)=∑1≤i<j≤n+1K̂ij.
The relative null spae of M at p is defined by [8] Np={X∈TpM∣ĥ(X,Y)=0,∀Y∈TpM}.

Theorem 4.1.

Let M̃(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,

for each unit vector X∈TpM orthogonal to ξ, we have
4Riĉ(X)≤(n+1)2‖Ĥ‖2+(n-1)(c+3)+(3c+13)‖TX‖2,

if Ĥ(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.

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) Let X∈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
(n+1)2‖Ĥ‖2=2τ̂+‖ĥ‖2-n(n-1)c+34-3c+134‖T‖2.
From (4.7), we get
(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
K̂ij=∑r=n+2m[ĥiirĥjjr-(ĥijr)2]+c+34+3c+134g2(ei,Tej),
and consequently
∑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
(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,
(n+1)22‖Ĥ‖2≥2Riĉ(X)-(n-1)(c+3)2-3c+132‖TX‖2,
which is equivalent to (4.6)

(ii) Assume that Ĥ(p)=0. Equality holds in (4.6) if and only if
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
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.

Let M be an n+1-dimensional invariant submanifold of a Sasakian space form M̃(c). Then,

for each unit vector X∈TpM orthogonal to ξ, we have
4Riĉ(X)≤(n-1)(c+3)+(3c+13).

A unit tanget vector X∈TpM orthogonal to ξ satisfies the equality case of (4.6) if and only if X∈𝒩p.

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.

ChenB.-Y.A Riemannian invariant for submanifolds in space forms and its applicationsBlairD. E.TanakaN.On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connectionsWebsterS. M.Pseudo-Hermitian structures on a real hypersurfaceTannoS.Variational problems on contact Riemannian manifoldsChenB.-Y.Some pinching and classification theorems for minimal submanifoldsChenB.-Y.Mean curvature and shape operator of isometric immersions in real-space-formsChenB.-Y.Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions