A Basic Inequality for the Tanaka-Webster Connection

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 (cid:2) namely, its pseudosectional curvature and pseudosectional curvature on one side (cid:3) and the main extrinsic invariant (cid:2) namely, squared pseudomean curvature on the other side (cid:3) with respect to the Tanaka-Webster connection. Moreover, involving the pseudo-Ricci curvature and the squared pseudomean 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. 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.


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.
for any X ∈ T M, where ∇ is the Levi-Civita connection of M. The structure of M 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 for all vector fields X and Y . Every Sasakian manifold is a K-contact manifold. Given a Sasakian manifold M, a plane section π in T p M 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 K π of a ϕ-section π is called ϕ-sectional curvature. If a Sasakian manifold M has constant ϕ-sectional curvature c, M is called a Sasakian space form, denoted by M c . For more details, see 2 .
Now let M be a submanifold immersed in M, 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 A v the Weingarten endomorphism associated with any v ∈ T ⊥ M. We put h r ij g h e i , e j , e r for any orthonormal vector e i , e j ∈ TM and e r ∈ T ⊥ M. The mean curvature vector field H is defined by H 1/ dim M 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 M 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. 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.

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 Also, by using 2.1 and 2.3 , we can see that We define the Tanaka-Webster curvature tensor of R in terms of ∇ by for all vector fields X, Y , and Z in M.
Let M c be a Sasakian space form of constant sectional curvature c and M a submanifold of M c . Then, we have the following Gauss' equation: for any tangent vector fields X, Y, Z tangent to M.
for any X, Y ∈ Γ TM , where h is called the lightlike second fundamental form of M with respect to the induced connection • ∇. In the view of 3.2 and 3.6 , From 3.7 , we obtain where ϕX TX NX. From 3.3 , 3.8 , and 3.9 it is easy to verify the following: Moreover, for the induced connection ∇, we have the following From the definition of R, together with 3.5 , we have

3.15
The pseudomean curvature vector field H is defined by H 1/ dim M trace h . M is said to be totally pseudogeodesic if the second fundamental h 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
We now recall the Chen's lemma.

3.17
Then, 2a 1 a 2 ≥ c, with the equality holding if and only if a 1 a 2 a 3 · · · a n .
Let p ∈ M and let π be a plane section of T p M which is generated by orthonormal vectors X and Y . We can define a function α π of tangent space T p M into 0, 1 by which is well defined. Now, we prove the following.

3.24
On the other hand, from 3.12 , we have  We now define a well-defined function δ M on M by using inf K p inf{ K π | π is a plane section ⊂ T p M} in the following manner: 3.30 If c −13/3, then we obtain directly from 3.19 the following result.

4.14
Since h e i , e n 1 ξ 0 from 3.10 , p is a totally pseudogeodesic point, and, hence, ϕ T p M ⊂ T p M. The converse is trivial. 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.