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Chen-Ricci inequality for submanifolds of contact metric manifolds.

1. INTRODUCTION

It is well known that Nash's immersion theorem [37] takes guarantee of an isometric immersion of every n-dimensional Riemannian manifold into the Euclidean space [E.sup.n(n+1)(3n+11)/2], and this phenomenon provides a natural motivation for the study of submanifolds of Riemannian manifolds. To find simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold is one of the basic interests in the submanifold theory. The main extrinsic invariant is the squared mean curvature and the main intrinsic invariants include the classical curvature invariants namely the Ricci curvature and the scalar curvature. There are also other important modern intrinsic invariants of (sub)manifolds introduced by B.-Y. Chen [16]. For a unit vector X in an n-dimensional submanifold M of a real space form [R.sup.m] (c), B.-Y. Chen [14, Theorem 4] obtained the following basic inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

involving the Ricci curvature Ric and the squared mean curvature [parallel]H[[parallel].sup.2] of the submanifold. The inequality (1.1) drew attention of several authors and they established same kind of inequalities for different kind of submanifolds in ambient manifolds possessing different kind of structures. The submanifolds included mainly invariant, anti-invariant and slant submanifolds [12], while ambient manifolds included mainly real, complex and Sasakian space forms (for example, see [15], [18], [25], [28], [29], [31], [32], [33], [34], [35], [46], [48], [50]). In search of a general theory, it is natural to ask about a basic inequality (corresponding to the inequality (1.1)) involving the Ricci curvature and the squared mean curvature of any submanifold of any Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. This goal was achieved in [22] (see Theorem 3.1) by exploiting the typical concept of k-Ricci curvature ([14], [52]) and the method of [14, Theorem 4].

On the other hand, the roots of contact geometry lie in differential equations as in 1872 Sophus Lie introduced the notion of contact transformation as a geometric tool to study systems of differential equations (for more details see [1], [2], [6] and [19]). In fact, the theory of contact metric structures occupies one of the leading places in researches of modern differential geometry because of its several applications in physics such as mechanics, optics, phase space of a dynamical system, thermodynamics and control theory ([1], [2]) and in the theory of geometrical quantization [24]. Furthermore, the internal contents of the theory of contact metric structures are very rich [6] and have close substantial interactions with other parts of geometry.

It is well known [40] that the tangent sphere bundle [T.sub.1][??] of a Riemannian manifold [??] admits a contact metric structure ([phi], [xi], [eta], [??]). If [??] is of constant curvature c = 1 then [T.sub.1][??] is Sasakian [45], that is, its curvature tensor [??] satisfies

[??] (X, Y)[xi] = [eta](Y)X ? [eta](X)Y, X, Y [member of] T[??]. (1.2) If c = 0 then the curvature tensor [??] satisfies [5]

[??] (X, Y)[xi] = 0. (1.3)

As a generalization of these two cases, in [7], D. E. Blair, T. Koufogiorgos and B. J. Papantoniou initiated the study of the class of contact metric manifolds [??], which satisfy

[??] (X, Y)[xi] = ([kappa]I + [mu]h) ([eta] (Y)X ? [eta] (X) Y), X,Y [member of] T[??], (1.4)

where [kappa], [mu] are real constants and the (1, 1)-tensor field h is defined by half of the Lie derivative of [phi] in the characteristic direction [xi]. A contact metric manifold belonging to this class is called a ([kappa], [mu])-manifold. Such a structure was first obtained by T. Koufogiorgos [26] by applying a D-homothetic deformation on a contact metric manifold satisfying (1.3), where for a positive constant a, by a D-homothetic deformation [43] one means a change of structure tensors of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

There is a complete classification of ([kappa], [mu])-manifolds given by E. Boeckx [10]. In non-Sasakian ([kappa], [mu])-manifolds the condition (1.4) determines the curvature completely; moreover, while the values of [kappa] and [mu] change, the form of (1.4) is invariant under D-homothetic deformations [7]. Any non-Sasakian ([kappa], [mu])-manifold is locally homogeneous and strongly locally [phi]-symmetric ([9], [11]). Characteristic examples of non-Sasakian ([kappa], [mu])-manifolds are the tangent sphere bundles of Riemannian manifolds of constant sectional curvature not equal to one and certain Lie groups [10]. If a ([kappa], [mu])-manifold has constant [phi]-sectional curvature c then it is called a ([kappa], [mu])-space form.

Thus motivated sufficiently, we study Ricci curvature and scalar curvature of submanifolds in contact metric manifolds. To be more specific, the results obtained (in this paper) for submanifolds of ([kappa], [mu])-space forms generalize the corresponding results for submanifolds of Sasakian space forms and at the same time several results and examples obtained for submanifolds of non-Sasakian ([kappa], [mu])-manifolds and non-Sasakian ([kappa], [mu])-space forms are quite different in nature with the corresponding results for submanifolds of Sasakian space forms. The paper is organized as follows. Section 2 consists of a brief introduction to ([kappa], [mu])-manifolds, ([kappa], [mu])-space forms and non-Sasakian ([kappa], [mu])-manifolds. In section 3, we recall the notion of Ricci curvature, k-Ricci curvature, scalar curvature, normalized scalar curvature and give a brief account of submanifolds. Then we recall the result including a basic inequality for submanifolds of any Riemannian manifold, which involves Ricci curvature and squared mean curvature of the submanifold. We also give a simple proof and discuss the conditions under which the inequality becomes equality. Since, this kind of inequality for Ricci curvature of a submanifold in a real space form was first obtained by B.-Y. Chen [14, Theorem 4], we name this inequality as Chen-Ricci inequality. In section 4, we obtain Chen-Ricci inequalities for C-totally real submanifolds in ([kappa], [mu])-space forms and non-Sasakian ([kappa], [mu])-manifolds. In particular, we derive Chen-Ricci inequality for C-totally real submanifolds in Sasakian space forms. We also give examples of C-totally real submanifolds of Sasakian space forms and non-Sasakian ([kappa], [mu])-manifolds, which satisfy Chen-Ricci equality. In section 5, we establish Chen-Ricci inequalities for submanifolds tangent to the structure vector field in ([kappa], [mu])-space forms and non-Sasakian ([kappa], [mu])-manifolds. As applications of these inequalities, using a surprising fact that every invariant submanifold of a non-Sasakian ([kappa], [mu])-manifold is totally geodesic, we observe that invariant submanifolds of non-Sasakian ([kappa], [mu])-manifolds and non-Sasakian ([kappa], [mu])-space forms satisfy Chen-Ricci equality, which is an improvement of a result of [47], where the Chen-Ricci equality is obtained as an inequality and necessary and sufficient condition for equality case is obtained. In [23] it is proved that a totally geodesic invariant submanifold of a Sasakian space form [??] (c) is Einstein if and only if c = 1. Here, we find an obstruction for an n-dimensional invariant submanifold of a non-Sasakian ([kappa], [mu])-space form [??] (c) to be Einstein. In particular, we deduce that invariant submanifolds of the tangent sphere bundle of a Riemannian manifold of constant curvature c [not equal to] 1 and having constant [phi]-sectional curvature can not be Einstein. Next, we find Chen-Ricci inequality for anti-invariant submanifolds tangent to the structure vector field in non-Sasakian ([kappa], [mu])-manifolds. Contrary to a known result that there is no anti-invariant submanifold of a Sasakian space form tangent to the structure vector field, which satisfies the corresponding Chen-Ricci equality; we provide an example of anti-invariant submanifold tangent to the structure vector field in a non-Sasakian ([kappa], [mu])-manifold with [kappa] = 0 = [mu], which satisfies Chen-Ricci equality. In section 6, we find basic inequalities involving scalar curvature and squared mean curvature for C-totally real submanifolds and submanifolds tangent to the structure vector field [xi] of ([kappa], [mu])-space forms and non-Sasakian ([kappa], [mu])-manifolds.

2. ([kappa], [mu])-MANIFOLDS

A 1-form [eta] on a differentiable manifold [??] of odd dimension 2m + 1 (m [greater than or equal to] 1) is called a contact form if [eta] ^ [(d[eta]).sup.m] [not equal to] 0 everywhere on [??], and [??] equipped with a contact form is a contact manifold. Since rank of d[eta] is 2m on the Grassmann algebra [LAMBDA][T.sup.?.sub.p][??] at each point p [member of] [??], therefore there exists a unique global vector field [xi], called the characteristic vector field, such that [eta]([xi]) = 1, d[eta]([xi], *) = 0, and consequently [L.sub.[xi]][eta] = 0, [L.sub.[xi]]d[eta] = 0, where [L.sub.[xi]] denotes the Lie differentiation by [xi]. In 1953, S. S. Chern [17] proved that the structural group of a (2m + 1)-dimensional contact manifold can be reduced to U(m)x1. A (2m+1)-dimensional differentiable manifold [??] is called an almost contact manifold [20] if its structural group can be reduced to U(m)x1. Equivalently, there is an almost contact structure ([phi], [xi], [eta]) [39] consisting of a tensor field [phi] of type (1, 1), a vector field [xi], and a 1-form [eta] satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

First and one of the remaining three relations of (2.1) imply the other two relations of (2.1). An almost contact structure is normal [41] if the torsion tensor [[phi], [phi]]+2d[eta][cross product][xi], where [[phi], [phi]] is the Nijenhuis tensor of [phi], vanishes identically. Let [??] be a compatible Riemannian metric with ([phi], [xi], [eta]), that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

where <*,*> denotes the inner product of the metric [??]. Then, [??] becomes an almost contact metric manifold equipped with an almost contact metric structure ([phi], [xi], [eta], [??]). The equation (2.2) is equivalent to

<X,[phi]Y> = ?<[phi]X, Y> alongwith <X, [xi]> = [eta] (X). (2.3)

An almost contact metric structure becomes a contact metric structure if <X,[phi]Y> = d[eta](X, Y) for all X, Y [member of] T[??]. In a contact metric manifold [??], the (1, 1)-tensor field h defined by 2h = [L.sub.[xi]][phi], which is the Lie derivative of [phi] in the characteristic direction [xi], is symmetric and satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

A contact metric manifold is called a K-contact manifold if the characteristic vector field [xi] is a Killing vector field. An almost contact metric manifold is a K-contact manifold if and only if [??][xi] = ?[phi], where [??] is the Levi-Civita connection. A K-contact manifold is a contact metric manifold, while the converse is true if h = 0. In a K-contact manifold, the curvature tensor [??] satisfies ([[??].sub.X[phi]])Y = [??]([xi],X)Y for all X, Y [member of] T[??]. A normal contact metric manifold is a Sasakian manifold. An almost contact metric manifold is Sasakian if and only if [42]

([[??].sub.X[phi]])Y = <X, Y>[xi] ? [eta](Y)X, X, Y [member of] T[??]. (2.5)

Also, a contact metric manifold [??] is Sasakian if and only if the curvature tensor [??] satisfies the equation (1.2). A Sasakian manifold is always a K-contact manifold and the converse is true in the dimension three. Thus a 3-dimensional contact metric manifold is Sasakian if and only if h = 0.

A contact metric manifold is a ([kappa], [mu])-manifold [??] if it satisfies (1.4). In a ([kappa], [mu])-manifold [??], one has [7]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7)

for all X, Y [member of] T[??]. The Ricci operator [??] satisfies [??][xi] = 2m[kappa][xi], where dim([??]) = 2m+1. Moreover, [h.sup.2] = ([kappa] ? 1) [[phi].sup.2] and the eigenvalues of h are 0, [lambda] and ?[lambda], where [lambda] = [square root of (1 ? [kappa])]. The eigenspace relative to the eigenvalue 0 is {[xi]}. Moreover, for [kappa] [not equal to] 1, the subbundle D = ker([eta]) can be decomposed in the eigenspace distributions [D.sub.+] and D_ relative to the eigenvalues [lambda] and ?[lambda], respectively. These distributions are orthogonal to each other and have dimension m. In a ([kappa], [mu])-manifold it follows that [kappa] [less than or equal to] 1. In fact, for a ([kappa], [mu])-manifold, the conditions of being a Sasakian manifold, a K-contact manifold, [kappa] = 1 and h = 0 are all equivalent. If [mu] = 0, then a ([kappa], [mu])-manifold is called an N([kappa])-contact metric manifold [8]. The tangent sphere bundle [T.sub.1][??] of a Riemannian manifold [??] of constant curvature c is a ([kappa], [mu])-manifold with [kappa] = c(2 ? c) and [mu] = ?2c.

Like complex space forms in Hermitian geometry, in contact metric geometry we have the notion of manifolds with constant [phi]-sectional curvature. In an almost contact metric manifold, for a unit vector X orthogonal to [xi], the sectional curvature [??](X ^ [phi]X) is called a [phi]-sectional curvature. In [27], T. Koufogiorgos showed that in a ([kappa], [mu])-manifold ([??],[phi], [xi], [eta], g) of dimension > 3 if the [phi]-sectional curvature at a point p is independent of the [phi]-section at p, then it is constant. If the ([kappa], [mu])-manifold [??] has constant [phi]-sectional curvature c then it is called a ([kappa], [mu])-space form and is denoted by [??] (c). The Riemann curvature tensor [??] of [??] (c) is given by [27]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8)

for all X, Y,Z [member of] T[??]. For a non-Sasakian ([kappa], [mu])-manifold [??], its Riemann curvature tensor [??] is given by ([9],[10])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

for all vector fields X, Y,Z in [??] .

If [kappa] = 1 then a ([kappa], [mu])-space form [??] (c) reduces to a Sasakian space form [??] (c) [38] and (2.8) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

In [44], S. Tanno asked whether there exists a non-Sasakian contact metric manifold of constant [phi]-sectional curvature. In [27], T. Koufogiorgos answered this problem in affirmative. A 3-dimensional non-Sasakian ([kappa], [mu])-manifold has a constant [phi]-sectional curvature, but for higher dimension this is not in general true. A non- Sasakian ([kappa], [mu])-manifold is of constant [phi]-sectional curvature c if and only if [mu] = [kappa]+1; in this case c = ?2k ? 1. In particular, the tangent sphere bundle of a manifold of constant curvature c [not equal to] 1 has constant [phi]-sectional curvature c2 = 9 [+ or -] 4?5 if and only if c = 2 [+ or -] [square root of (5)]. For more details we refer to [6], [7] and [27].

3. CHEN-RICCI INEQUALITY FOR RIEMANNIAN SUBMANIFOLDS

Let M be an n-dimensional Riemannian manifold. Let {[e.sub.1], ..., [e.sub.k]}, 2 [less than or equal to] k [less than or equal to] n, be an orthonormal basis of a k-plane section [[PI].sub.k] of [T.sub.p]M. If k = n then [[PI].sub.n] = [T.sub.p]M; and if k = 2 then [[PI].sub.2] is a plane section of [T.sub.p]M. For a fixed i [member of] {1, ..., k}, a k-Ricci curvature of [[PI].sub.k] at [e.sup.i], denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([e.sub.i]), is defined by [14]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

where K ([e.sub.i] ^ [e.sub.j]) is the sectional curvature of the plane section spanned by [e.sub.i] and [e.sub.j]. An n-Ricci curvature [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the usual Ricci curvature of [e.sub.i], denoted Ric([e.sub.i]). Thus for any orthonormal basis {[e.sub.1], ..., [e.sub.n]} for [T.sub.p]M and for a fixed i [member of] {1, ..., n}, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The scalar curvature [tau]([[PI].sub.k]) of the k-plane section [[PI].sub.k] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

Geometrically, [tau]([[PI].sub.k]) is the scalar curvature of the image expp([[PI].sub.k]) of [[PI].sub.k] at p under the exponential map at p. We define the normalized scalar curvature [[tau].sub.N] ([[PI].sub.k]) of [[PI].sub.k] by [23]

[[tau].sub.N] ([[PI].sub.k]) = 2[tau]([[PI].sub.k])/k (k ? 1). (3.3)

The normalized scalar curvature at p is defined as [13]

[[tau].sub.N] (p) = 2[tau](p)/n (n ? 1). (3.4)

Then, we see that [[tau].sub.N] (p) = [[tau].sub.N] ([T.sub.p]M). The scalar curvature [tau] (p) of M at p is identical with the scalar curvature of the tangent space [T.sub.p]M of M at p, that is, [tau](p) = [tau]([T.sub.p]M). If [[PI].sub.2] is a plane section and {[e.sub.1], [e.sub.2]} is any orthonormal basis for [[PI].sub.2], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let M be a submanifold of a Riemannian manifold [??] equipped with a Riemannian metric [??]. Covariant derivatives and curvatures with respect to (M,g) will be written in the usual manner, while those with respect to the ambient manifold ([??], [??]) will be written with "tildes" over them. We use the inner product notation/// <*,*> for both the metrics [??] of [??] and the induced metric g on the submanifold M. The Gauss and Weingarten formulas are given respectively by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all X, Y [member of] TM and N [member of] [T.sup.[perpendicular to]]M, where [??], [nabla] and [[nabla].sup.[perpendicular to] are respectively the Riemannian, induced Riemannian and induced normal connections in [??], M and the normal bundle [T.sup.[perpendicular to]]M of M respectively, and [sigma] is the second fundamental form related to the shape operator A by <[sigma](X,Y),N> = <[A.sub.N]X,Y>. The equation of Gauss is given by

R(X,Y,Z,W) = [??](X,Y,Z,W) + <[sigma](X,W),[sigma](Y,Z)> ? <[sigma](X,Z),[sigma](Y,W)> (3.5)

for all X, Y,Z,W [member of] TM, where [??] and R are the curvature tensors of [??] and M respectively.

The mean curvature vector H is given by nH = trace([sigma]), where n = dim(M). The submanifold M is totally geodesic in [??] if [sigma] = 0, and minimal if H = 0. If [sigma] (X, Y) = g (X, Y)H for all X, Y [member of] TM, then M is totally umbilical.

Now, let {[e.sub.1], ... , [e.sub.n]} be an orthonormal basis of the tangent space [T.sub.p]M and [e.sub.r] belongs to an orthonormal basis {[e.sub.n]+1, ... , em} of the normal space [T.sup.[perpendicular to]]p M. We put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let K([e.sub.i] ^ [e.sub.j]) and [??] ([e.sub.i] ^ [e.sub.j]) denote the sectional curvature of the plane section spanned by [e.sub.i] and [e.sub.j] at p in the submanifold M and in the ambient manifold [??] respectively. In view of the equation (3.5) of Gauss, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

From (3.6) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.7)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

denote the scalar curvature of the n-plane section [T.sub.p]M in the ambient manifold [??]. Also, the squared second fundamental form and the squared mean curvature satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8)

The relative null space of M at p is defined by [14]

[N.sub.p] = {X [member of] [T.sub.p]M | [sigma](X, Y) = 0 for all Y [member of] [T.sub.p]M},

which is also known as the kernel of the second fundamental form at p [15]. We denote the set of unit vectors in [T.sub.p]M by [T.sup.1.sub.p]; thus

[T.sup.1.sub.p] = {X [member of] [T.sub.p]M | <X,X> = 1}.

Now, we recall the following result, as a general theory, in which a basic inequality involving the Ricci curvature and the squared mean curvature of any submanifold of a Riemannian manifold is established. From now on, the inequality (3.9) will be called the Chen-Ricci inequality. Equality case of Chen-Ricci inequality will be known as the Chen-Ricci equality.

Theorem 3.1. ([22, Theorem 3.1]) Let M be an n-dimensional submanifold of a Riemannian manifold. Then, the following statements are true.

(a) For X [member of] [T.sup.1.sub.p]M, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the n-Ricci curvature of [T.sub.p]M at X with respect to the ambient manifold [??].

(b) The equality case of (3.9) is satisfied by X [member of] [T.sup.1.sub.p]M if and only if

[sigma] (X,X) = n/2 H (p) and [sigma] (X, Y) = 0 (3.10)

for all Y [member of] [T.sub.p]M such that <X, Y> = 0.

(c) The equality case of (3.9) holds for all X [member of] [T.sup.1.sub.p]M if and only if either p is a totally geodesic point or n = 2 and p is an umbilical point.

Proof. We give a very simple proof taken from [21]. We put

2[sigma]' (X, Y) = [sigma] (X, Y) ? n/2<X,Y> H, X,Y [member of] [T.sub.p]M.

Then for a unit vector X [member of] [T.sub.p]M, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In view of the Gauss equation (3.5) and the above inequality, we can easily get our theorem.

From Theorem 3.1, we immediately have the following

Corollary 3.1. Let M be an n-dimensional submanifold of a Riemannian manifold. Then for any X [member of] [T.sup.1.sub.p]M any two of the following three statements imply the remaining one.

(a) The mean curvature vector H(p) vanishes.

(b) The unit vector X belongs to the relative null space [N.sub.p].

(c) The unit vector X satisfies the equality case of (3.9), namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.11)

4. CHEN-RICCI INEQUALITY FOR C-TOTALLY REAL SUBMANIFOLDS

A submanifold M of an almost contact metric manifold is said to be antiinvariant ([54], [55]) if [phi](TM) [subset or equal to] [T.sup.[perpendicular to]]M. A submanifold M in a contact manifold is called an integral submanifold [6] if every tangent vector of M belongs to the contact distribution defined by [eta] = 0. In this case we have d[eta] (X, Y) = 0, X, Y [member of] TM. An integral submanifold M in a contact metric manifold is called a C-totally real submanifold [53]. If a submanifold M in a contact metric manifold is normal to the structure vector field [xi], then it is anti-invariant. Thus C-totally real submanifolds in a contact metric manifold are anti-invariant, as they are normal to [xi].

For a submanifold M of an almost contact metric manifold, we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where PX, [h.sup.T]X and [([phi]h).sup.T]X are the tangential parts of [phi]X, hX and [phi]hX respectively for all X [member of] TM. If M is C-totally real then in view of

<[A.sub.[xi]]X, Y> = <?[[??].sub.X][xi],Y> = <[phi]X + [phi]hX, Y>,

it follows that [A.sub.[xi]] = [([phi]h).sup.T].

Let M be a submanifold of a ([kappa], [mu])-manifold [??]. If [??] is a non-Sasakian ([kappa], [mu])-manifold, we define the 1-form, denoted by [t.sub.2] on M by [47]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.1)

for all X [member of] TM. If [??] is a ([kappa], [mu])-space form, we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.2)

Now, we obtain Chen-Ricci inequality for a C-totally real submanifold in a ([kappa], [mu])-space form.

Theorem 4.1. Let M be an n-dimensional C-totally real submanifold of a contact metric manifold [??]. Then, the following statements are true.

(a) If [??] is a ([kappa], [mu])-space form [??] (c) then the Chen-Ricci inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

for every X [member of] [T.sup.1.sub.p]M.

(b) If [??] is a non-Sasakian ([kappa], [mu])-manifold then the Chen-Ricci inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

for every X [member of] [T.sup.1.sub.p]M.

(c) A unit vector X [member of] [T.sub.p]M satisfies the equality case of (4.3) (resp. (4.4)) if and only if (3.10) is true. In particular, if H(p) = 0, then a unit vector X [member of] [T.sub.p]M satisfies the equality case of (4.3) (resp. (4.4)) if and only if X belongs to the relative null space [N.sub.p].

(d) The equality case of (4.3) (resp. (4.4)) is true for all unit vectors in [T.sub.p]M if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.

Proof. Let {[e.sub.1], ... , [e.sub.n]} be an orthonormal basis of the tangent space [T.sub.p]M. Then from (2.8) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.5)

From (4.5) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.6)

where (4.2) is also used. Using (4.6) in (3.9) we find the Chen-Ricci inequality (4.3). Similarly, in view of (2.9) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.7)

From (4.7) and (4.1) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.8)

Finally the inequality (4.4) is obtained by using (4.8) in (3.9). Rest of the proof is straightforward.

Putting h = 0 in (4.3), we have the following

Theorem 4.2. If M is an n-dimensional C-totally real submanifold of a Sasakian space form [??] (c), then the following statements are true.

(a) It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.9)

(b) The equality case of (4.9) is satisfied by X [member of] [T.sup.1.sub.p]M if and only if (3.10) is true. If H(p) = 0, X [member of] [T.sup.1.sub.p]M satisfies equality in (4.9) if and only if X [member of] [N.sub.p].

(c) The equality case of (4.9) holds for all X [member of] [T.sup.1.sub.p]M if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.

By polarization, from Theorem 4.2, we derive

Theorem 4.3. Let M be an n-dimensional C-totally real submanifold of a Sasakian space form [??] (c). Then

S [less than or equal to] 1/4 {[n.sup.2][parallel]H[[parallel].sup.2] + (n ? 1)(c + 3)}g. (4.10)

with the equality case holding identically if and only if either M is totally geodesic submanifold or M is a totally umbilical surface.

Remark 4.1. The Chen-Ricci inequality (4.3) is same as the inequality (33) in [49] but is different from the inequality (3.1) in [3, Theorem 3.1]. Moreover, the assumption of the statement in [3, Corollary 3.2] is also not true, because the submanifold M in a contact metric manifold normal to the structure vector [xi] can not be invariant. The inequality (4.9) is the same as the inequalities (5.1) of Theorem 5.1 in [23], (i) of Theorem 1 in [31], (2.1) of Theorem 2.1 in [35] and (2.1) in Theorem 2.1 in [32]. The inequality (4.10) is same as the inequalities (5.3) of Theorem 5.2 in [23], (9) in the Theorem 3.1 in [29] or [28], (2.9) in Theorem 2.2 in [32] and the inequality in Theorem 2 in [31]).

Now, we give examples of C-totally real submanifolds of Sasakian space forms and non-Sasakian ([kappa], [mu])-manifolds, which satisfy Chen-Ricci equality.

Example 4.1. An n-dimensional compact minimal C-totally real submanifold of a Sasakian space form [M.sup.2n+1] (c), c > ?3 with positive sectional curvature is totally geodesic [51, Theorem 4]; thus it is an Einstein manifold and satisfies 4S = (n ? 1)(c + 3)g.

Example 4.2. Given a point p of a non-Sasakian ([kappa], [mu])-manifold, there are at least two C-totally real submanifolds, passing through p. To see this, consider the foliations given by the eigendistributions of h; then their leaves are C-totally real submanifolds of the given non-Sasakian ([kappa], [mu])-manifold. These submanifolds are totally geodesic, therefore the equality case of (4.4) is satisfied.

5. CHEN-RICCI INEQUALITY FOR SUBMANIFOLDS TANGENT TO THE STRUCTURE VECTOR FIELD [xi]

Let M be a submanifold of a contact metric manifold such that the structure vector field [xi] is tangent to M. Then from (2.4) we get

0 = ?[phi][xi] ? [phi]h[xi] = [[??].sub.[xi]][xi] = [[??}.sub.[xi]][xi] + [sigma] ([xi],[xi]),

which gives

[[??].sub.[xi]][xi] = 0 and [sigma] ([xi], [xi]) = 0. (5.1)

Now, if M is also totally umbilical then H = <[xi], [xi]> H = [sigma]([xi], [xi]) = 0 and we have the Following

Lemma 5.1. Every totally umbilical submanifold of a contact metric manifold, such that the structure vector field of the ambient manifold is tangent to the submanifold, is minimal and consequently totally geodesic.

In the following Theorem, we find Chen-Ricci inequality for submanifolds tangent to the structure vector field in a ([kappa], [mu])-manifold.

Theorem 5.1. Let M be an n-dimensional submanifold of a contact metric manifold [??] such that [xi] [member of] TM. Then the following statements are true:

(a) If [??] is a ([kappa], [mu])-space form [??] (c) then the Chen-Ricci inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.2)

for all unit vectors X [member of] [T.sup.1.sub.p]M.

(b) If [??] is a non-Sasakian ([kappa], [mu])-manifold then the Chen-Ricci inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.3)

where X is any unit vector in [T.sup.1.sub.p]M.

(c) A unit vector X [member of] [T.sub.p]M satisfies the equality case of (5.2) (resp. (5.3)) if and only if (3.10) is true. In particular, if H(p) = 0, then a unit vector X [member of] [T.sub.p]M satisfies the equality case of (5.2) (resp. (5.3)) if and only if X belongs to the relative null space Np.

(d) The equality in (5.2) (resp. (5.3)) holds identically for all unit tangent vectors in [T.sub.p]M if and only if p is a totally geodesic point.

Proof. LetM be an n-dimensional submanifold of a ([kappa], [mu])-space form [??] (c) such that the structure vector field [xi] is tangent to M. Let {[e.sub.1], ... , [e.sub.n]} be an orthonormal basis of the tangent space [T.sub.p]M. From (2.8) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.4)

From (5.4) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.5)

Using (5.5) in (3.9) we find the Chen-Ricci inequality (5.2).

From (2.9) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.6)

which gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.7)

Using (5.7) in (3.9) we find the Chen-Ricci inequality (5.3). The proof of (c) is straightforward. Since [xi] [member of] TM, therefore from Lemma 5.1, each umbilical point is a totally geodesic point; thus (d) is correct.

An n-dimensional submanifold M of an almost contact metric manifold [??] is said to be invariant [55] if [phi](TM) ? TM. If [??] is a contact metric manifold equipped with the contact metric structure ([phi], [xi], [eta], [??]), then the structure vector field [xi] becomes tangent to the invariant submanifold M, [sigma] (X, [xi]) = 0 and M is minimal [6]. Choosing an orthonormal basis {[e.sub.i], [phi][e.sub.i], [xi]}, i = 1, ..., (n ? 1) /2, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.8)

In general, an invariant submanifold of a Sasakian manifold needs not to be totally geodesic. For example, the circle bundle (S,[Q.sup.n]) over an n-dimensional complex quadric Qn in a complex projective space [C.sup.Pn+1] is an invariant submanifold of a (2n + 3)-dimensional Sasakian space form [S.sup.2n+3(c)] with c > ?3, which is not totally geodesic [55, pp. 328-329]. But, in non-Sasakian case, we have the following useful result.

Theorem 5.2. [36, Theorem 3.1] Every invariant submanifold of a non-Sasakian ([kappa], [mu])-manifold is totally geodesic.

In view of Theorems 5.1 and 5.2 we have the following

Theorem 5.3. For an n-dimensional invariant submanifold M in a contact metric manifold [??] the following statements are true:

(a) If [??] is a non-Sasakian ([kappa], [mu])-manifold then the invariant submanifold M satisfies Chen-Ricci equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.9)

for each unit vector X [member of] [T.sub.p]M.

(b) If [??] is a non-Sasakian ([kappa], [mu])-space form [??] (c) then the invariant submanifold M satisfies Chen-Ricci equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.10)

for each unit vector X [member of] [T.sub.p]M.

Remark 5.1. The part (a) of the above Theorem is an improvement of Theorem 7.4 of [47], where the Chen-Ricci equality (5.9) is obtained as an inequality and necessary and sufficient condition for equality case is given.

In Theorem 4.5 of [23], it is proved that a totally geodesic invariant submanifold of a Sasakian space form [??] (c) is Einstein if and only if c = 1. Here, we prove the following

Proposition 5.1. Let M be an n-dimensional invariant submanifold of a non- Sasakian ([kappa], [mu])-space form [??] (c). If M is Einstein, then

c = ? 1/3 (5 ? 8/n ? 1). (5.11)

Proof. Let M be an n-dimensional invariant submanifold of a non-Sasakian ([kappa], [mu])- space form [??] (c). Suppose that M is Einstein. Let X be a unit vector in [T.sup.1.sub.p]M?{[xi]} orthogonal to [xi]. Then from (5.10) it follows that

0 = Ric(X)?Ric([xi]) = ([mu] + n ? 3) <[h.sup.T]X,X> + 1/4((n+ 1) c + 3(n ? 3) ? 4 (n ? 2) [kappa]) .

Since trace ([h.sup.T]) = 0, from the previous equation we get

0 = (n + 1) c + 3(n ? 3) ? 4 (n ? 2) [kappa].

Using c = ?2[kappa] ? 1 from the above equation we get (5.11).

As an important deduction, we have the following

Theorem 5.4. Invariant submanifolds of the tangent sphere bundle of a Riemannian manifold of constant curvature c [not equal to] 1 and having constant [phi]-sectional curvature can not be Einstein.

Proof. It is known that the tangent sphere bundle [??] of a manifold of constant curvature c [not equal to] 1 has constant [phi]-sectional curvature [c.sup.2] = 9 [+ or -] 4[square root of (5)] if and only if c = 2 [+ or -] [square root of (5)] [27]. Now, let M be an invariant submanifold of the tangent sphere bundle [??] . If M is Einstein then in view of Proposition 5.1, we have

9 [+ or -] 4[square root of (5)] = ?1/3 (5 ? 8/n ? 1),

which has no integer solution for n.

Now, we consider the anti-invariant submanifolds of contact metric manifolds tangent to the structure vector field [xi]. Putting P = 0 in the Chen-Ricci inequality (5.3), we immediately get the following

Corollary 5.1. Let M be an n-dimensional anti-invariant submanifold in a non- Sasakian ([kappa], [mu])-manifold such that [xi] [member of] TM. Then:

(a) The Chen-Ricci inequality becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.12)

where X is any unit vector in [T.sub.p]M.

(b) The unit vector X [member of] [T.sub.p]M satisfies the equality case of (5.12) if and only if (3.10) is true. In particular, if H(p) = 0, then the unit vector X [member of] [T.sub.p]M satisfies the equality case of (5.12) if and only if X belongs to the relative null space [N.sub.p].

(c) The equality in (5.12) holds identically for all unit tangent vectors in [T.sub.p]M if and only if p is a totally geodesic point.

Now, we recall the following result.

Theorem 5.5. (Theorem 4.12 (b), [23]) Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form [??](c) tangent to [xi]. If X [member of] [T.sub.p]M is a unit vector perpendicular to [xi] then

Ric(X) < 1/4 {[n.sup.2][parallel]H[[parallel].sup.2] + (n ? 1)(c + 3) ? (c ? 1)}. (5.13)

From the previous result, it is clear that an anti-invariant submanifold of a Sasakian space form tangent to the structure vector field can not satisfy the Chen-Ricci equality for arbitrary unit vectors. In contrast to this fact, we give the following

Example 5.1. Let ([??],[phi], [xi], [eta], g) be a contact metric manifold satisfying [??](X, Y)[xi] = 0 for all X, Y [member of] T[??]. These manifolds have been deeply studied by D. Blair in [5], where he proves that the distribution [D.sub.+] [direct sum] {[xi]} is integrable and defines a totally geodesic foliation of M, where [D.sub.+] is the eigenspace distribution corresponding to the eigenvalue [lambda] = 1 of h. Thus the leaves of [D.sub.+] [direct sum] {[xi]} give examples of totally geodesic submanifolds of [??], which are anti-invariant because [phi][D.sub.+] = [D.sub.?] so that [phi] maps tangent vectors into normal vectors. The unit vectors in these leaves satisfy the Chen-Ricci equality (the equality case of (5.12)), namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.14)

where [kappa] = [mu] = [parallel]H[parallel] = 0 has been used in (5.12). In particular, Ric ([xi]) = 0.

6. SCALAR CURVATURE OF SUBMANIFOLDS

In view of (3.7) it follows that an n-dimensional submanifold M of a Riemannian manifold satisfies

[tau] (p) [less than or equal to] 1/2 [n.sup.2][parallel]H[[parallel].sup.2] + [tau]([T.sub.p]M) (6.1)

at each point p [member of] M with equality if and only if p is a totally geodesic point. The improved version of the inequality (6.1) is given in the following:

Theorem 6.1. ([21, Theorem 4.2]) For an n-dimensional submanifold M in an m-dimensional Riemannian manifold, at each point p [member of] M, we have

[tau] (p) [less than or equal to] n (n ? 1)/2 [parallel]H[[parallel].sup.2] + [??]([T.sub.p]M) (6.2)

or equivalently

[[tau].sub.N](p) [less than or equal to] [parallel]H[[parallel].sup.2] + [[??].sub.N] ([T.sub.p]M) (6.3)

with equality if and only if p is a totally umbilical point.

Definition 6.1. The inequality (6.2) or (6.3) will be known as the scalar inequality.

Let M be a submanifold of a ([kappa], [mu])-manifold [??]. If [??] is a non-Sasakian ([kappa], [mu])-manifold, we write [47]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.4)

If [??] is a ([kappa], [mu])-space form, we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.5)

Now, we obtain scalar inequality for submanifolds of ([kappa], [mu])-manifolds. In fact, we have the following

Theorem 6.2. Let M be an n-dimensional C-totally real submanifold of a contact metric manifold [??]. Then the following statements are true.

(a) If [??] is a ([kappa], [mu])-space form [??] (c) then the scalar inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.6)

(b) If [??] is a non-Sasakian ([kappa], [mu])-manifold then the scalar inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.7)

(c) The equality cases of (6.6) (resp. (6.7)) are true if and only if p is a totally umbilical point.

Proof. From (4.6) and (6.5) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.8)

which in view of (6.2) gives (6.6). Similarly, from (4.8) and (6.4) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.9)

Using (6.9) in (6.2) gives (6.7). _

Using h = 0, we immediately have the following

Corollary 6.1. Let M be an n-dimensional C-totally real submanifold of a Sasakian space form [??] (c). Then the scalar inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.10)

The equality cases of (6.10) is true if and only if p is a totally umbilical point.

Next, we have the following

Theorem 6.3. Let M be an n-dimensional submanifold of a contact metric manifold [??] tangent to the structure vector field [xi]. Then the following statements are true.

(a) If [??] is a ([kappa], [mu])-space form [??] (c) then the scalar inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.11)

(b) If [??] is a non-Sasakian ([kappa], [mu])-manifold then the scalar inequality is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.12)

(c) The equality cases of (6.11) (resp. (6.12)) are true if and only if p is a totally umbilical point.

Proof. Using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]([e.sub.i]) from (5.5) and (6.5), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.13)

In view of (6.13) and (6.2) we get (6.11). Similarly, from (5.7) and (6.4) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.14)

Using (6.14) in (6.2) one obtains (6.12).

The inequality (6.12) is same as (3.8) of Theorem 3.3 in [47].

In view of Theorems 6.3 and 5.2 and the relations (5.8) we have the following

Theorem 6.4. For an n-dimensional invariant submanifold M in a contact metric manifold [??] the following statements are true:

(a) If [??] is a non-Sasakian ([kappa], [mu])-space form [??] (c) then

[tau] = 1/8(n ? 1) ((n + 1) c + 8[kappa] + 3(n ? 3)). (6.15)

(b) If [??] is a non-Sasakian ([kappa], [mu])-manifold then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.16)

Remark 6.1. The part (b) of the above Theorem is an improvement of Theorem 7.1 of [47], where the scalar equality (6.16) is obtained as an inequality and a necessary and sufficient condition for equality case is given.

Dedicated to the 65th birthday of Professor Bang-Yen Chen

Received: December 23, 2008.

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MUKUT MANI TRIPATHI

Banaras Hindu University

Faculty of Science

Department of Mathematics

Varanasi 221005, India

E-mail address: mmtripathi66@yahoo.com
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