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 (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.
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  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.
Let be an odd-dimensional Riemannian manifold with a Riemannian metric satisfying
Then is called the almost contact metric structure on . Let denote the fundamental 2 form in given by for all , the set of vector fields of . If , then is said to be a contact metric manifold. Moreover, if is a Killing vector field with respect to , and the contact metric structure is called a -contact structure. Recall that a contact metric manifold is -contact if and only if for any , where is the Levi-Civita connection of . The structure of is said to be normal if , where is the Nijenhuis torsion of . 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 and . Every Sasakian manifold is a -contact manifold.
Given a Sasakian manifold , a plane section in is called a -section if it is spanned by and , where is a unit tangent vector field orthogonal to . The sectional curvature of a -section is called -sectional curvature. If a Sasakian manifold has constant -sectional curvature is called a Sasakian space form, denoted by . (For more details, see ).
Now let be a submanifold immersed in . We also denote by the induced metric on . Let be the Lie algebra of vector fields in and the set of all vector fields normal to . We denote by the second fundamental form of and by the Weingarten endomorphism associated with any . We put for any orthonormal vector , and . The mean curvature vector field is defined by . is said to be totally geodesic if the second fundamental form vanishes identically.
From now on, we assume that the dimension of is , and that of the ambient manifold is . We also assume that the structure vector field is tangent to . Hence, if we denote by the orthogonal distribution to in , we have the orthogonal direct decomposition of by . For any , we write , where (, resp.) is the tangential (normal, resp.) component of . If is a -contact manifold, (2.2) gives for any in . Given a local orthonormal frame of , we can define the squared norms of and by resepectively. It is easy to show that both and are independent of the choice of the orthonormal frames. The submanifold is said to be invariant if is identically zero, that is, for any . On the other hand, is said to be an anti-invariant submanifold if is identically zero, that is, for any .
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  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 for all vector fields . Together with (2.1), is written by Also, by using (2.1) and (2.3), we can see that
We define the Tanaka-Webster curvature tensor of (in terms of ) by for all vector fields , and in .
Let be a Sasakian space form of constant sectional curvature and a submanifold of . Then, we have the following Gauss’ equation: for any tangent vector fields tangent to .
Let us define the connection on induced from the Tanaka-Webster connection on given by for any , where is called the lightlike second fundamental form of with respect to the induced connection . In the view of (3.2) and (3.6), From (3.7), we obtain where .
For an orthonormal basis of the tangent space , the pseudoscalar curvature at is defined by where denotes the pseudosectional curvature of associated with the plane section spanned by and for the Tanaka-Webster connection . In particular, if we put , then (3.13) implies that Moreover, from (3.9), we have The pseudomean curvature vector field is defined by . is said to be totally pseudogeodesic if the second fundamental 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 ,
We now recall the Chen's lemma.
Lemma 3.1 (see ). Let be real numbers such that Then, , with the equality holding if and only if .
Let and let be a plane section of which is generated by orthonormal vectors and . We can define a function of tangent space into by which is well defined.
Now, we prove the following.
Theorem 3.2. Let be an -dimensional submanifold isometrically immersed in a -dimensional Sasakian space form such that the structure vector field is tangent to in terms of the Tanaka-Wester connection . Then, for each point and each plane section , we have the following: Equality in (3.19) holds at if and only if there exist an orthonormal basis of and an orthonormal basis of such that and the shape operators , , take the following forms:
Proof. Let be a submanifold of . We introduce Then, from (3.16) and (3.21), we get Let be a point of and let be a plane section at . We choose an orthonormal basis for and for such that , Span, and the pseudomean curvature vector is parallel to . Then, from (3.22), we get and so, by applying Lemma 3.1, we obtain On the other hand, from (3.12), we have Then, from (3.24) and (3.25), we get 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 Moreover, choosing and such that , from (3.11), we also have the following Thus, with respect to the chosen orthonormal basis , the shape operators of take the forms.
We now define a well-defined function on by using in the following manner: If , then we obtain directly from (3.19) the following result.
Corollary 3.3. Let be an -dimensional submanifold isometrically immersed in a -dimensional Sasakian space form such that the structure vector field is tangent to in terms of the Tanaka-Wester connection . Then, for each point and each plane section , we have the following: The equality in (3.31) holds if and only if is a anti-invariant submanifold with .
Proof. In order to estimate , we minimize in (3.19). For an orthonormal basis of with , we write Thus, we see that the minimum value of is zero, provided that is orthogonal to , and is orthogonal to . Thus we have (3.31) with equality case holding if and only if is anti-invariant such that .
4. A Pseudo-Ricci Curvature for Sasakian Space Form
We denote the set of unit vectors in by by Let , , be an orthonormal basis of a -place section of . If , then , and if , then is a plane section of . For a fixed , a -pseudo-Ricci curvature of at , denoted by , is defined by  where is the pseudosectional curvature in terms of the Tanaka-Webster connection of the plane section spanned by and . We note that an -pseudo-Ricci curvature is the usual pseudo-Ricci curvature of , denoted by . Thus, for any orthonormal basis for and for a fixed , we have the following: The pseudoscalar curvature of the -plane section is given by The relative null spae of at is defined by 
Theorem 4.1. Let be a -dimensional Sasakian space form and an -dimensional submanifold tangent to with respect to the Tanaka-Webster connection . Then, (i)for each unit vector orthogonal to , we have (ii)if , then a unit tanget vector orthogonal to satisfies the equality case of (4.6) if and only of . (iii)the equality case of (4.6) holds identically for all unit tangent vectors orthogonal to at if and only if is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.
Proof. (i) Let be a unit tangent vector at , orthogonal to . We choose an orthonormal basis for and for such that and . Then, from (3.16), we have
From (4.7), we get
From (3.12), we have
Substituting (4.10) into (4.8), one gets
which is equivalent to (4.6)
(ii) Assume that . Equality holds in (4.6) if and only if Then, for each , that is, .
(iii) The equality case of (4.6) holds for all unit tangent vectors at if and only if Since from (3.10), is a totally pseudogeodesic point, and, hence, . The converse is trivial.
Corollary 4.2. Let be an -dimensional invariant submanifold of a Sasakian space form . Then, (i)for each unit vector orthogonal to , we have (ii)A unit tanget vector orthogonal to satisfies the equality case of (4.6) if and only if . (iii)The equality case of (4.6) holds identically for all unit tangent vectors orthogonal to at if and only if is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.
D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 203, Birkhäuser Boston, Boston, Mass, USA, 2002.View at: Zentralblatt MATH