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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 798317, 7 pages
http://dx.doi.org/10.1155/2008/798317
Research Article

Harmonic Maps and Stability on -Kenmotsu Manifolds

Universitá degli Studi di Parma, Viale G.P. Usberti 53-A, Parma 43100, Italy

Received 13 June 2007; Accepted 9 January 2008

Academic Editor: Mircea-Eugen Craioveanu

Copyright © 2008 Vittorio Mangione. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to study some submanifolds and Riemannian submersions on an -Kenmotsu manifold. The stability of a -holomorphic map from a compact -Kenmotsu manifold to a Kählerian manifold is proven.

1. Introduction

In Section 2, we give preliminaries on -Kenmotsu manifolds. The concept of -Kenmotsu manifold, where is a real constant, appears for the first time in the paper of Jannsens and Vanhecke [1]. More recently, Olszak and Roşca [2] defined and studied the -Kenmotsu manifold by the formula (2.3), where is a function on such that . Here, is the dual -form corresponding to the characteristic vector field of an almost contact metric structure on . The condition follows in fact from (2.3) if . This does not hold in general if .

A -Kenmotsu manifold is a Kenmotsu manifold (see Kenmotsu [3, 4]. Theorem 2.1 provides a geometric interpretation of an -Kenmotsu structure.

In Section 3, we initiate a study of harmonic maps when the domain is a compact -Kenmotsu manifold and the target is a Kähler manifold.

Ianus and Pastore [5, 6] defined a -holomorphic map between an almost contact metric manifold and an almost Hermitian manifold as a smooth map such that the condition is satisfied. Then, the formula holds, where is the tension field of and , being the connection induced in the pull-back bundle (see [7]). It is easy to see that in our assumptions and so that a -holomorphic map between an -Kenmotsu manifold and a Kähler manifold is a harmonic map. If is a compact manifold, a second-order elliptic operator , called the Jacobi operator, is associated to the harmonic map . It is well known that the spectrum of consists only of a discrete set of an infinite number of eigenvalues with finite multiplicities, bounded by the first one. We define the Morse index of the harmonic map as the sum of multiplicities of negative eigenvalues of the Jacobi operator [8, 9]. A harmonic map is called stable if the Morse index is zero. We have proven that any -holomorphic map from a compact -Kenmotsu manifold to a Kähler manifold is a stable harmonic map (see [10]).

2. -Kenmotsu Manifolds

A differentiable -dimensional manifold is said to have a -structure or an almost contact structure if there exist a tensor field of type (1, 1), a vector field , and a -form on satisfying where denotes the identity transformation.

It seems natural to include also and ; both can be derived from (2.1).

Let be an associated Riemannian metric on such that Putting in (2.2) and using (2.1), we get , for any vector field on .

In this paper, we denote by and the algebra of smooth functions on and the -module of smooth sections of a vector bundle , respectively. All manifolds are assumed to be connected and of class . Tensors fields, distribution, and so on are assumed to be of class if not stated otherwise.

We say that is an -Kenmotsu manifold if there exists an almost contact metric structure on satisfying for , where is a smooth function on such that .

A -Kenmotsu manifold is a Kenmotsu manifold [2, 3].

The following theorem provides a geometric interpretation of any -Kenmotsu structure.

Theorem 2.1 (Olszak-Roşca). Let be an almost contact metric manifold. Then, is -Kenmotsu if and only if it satisfies the following conditions:
(a) the distribution is integrable and any leaf of the foliation corresponding to is a totally umbilical hypersurface with constant mean curvature; (b) the almost Hermitian structure induced on an arbitrary leaf is Kähler; (c) and .

Moreover, we have which gives .

The characteristic vector field of an -Kenmotsu manifold also satisfies

Levy proven that a second-order symmetric parallel nonsingular tensor on a space of constant curvature is a constant multiple of the metric tensor [11]. On the other hand, Sharma proven that there is no nonzero skew-symmetric second-order parallel tensor on a Sasakian manifold [12]. For an -Kenmotsu manifold we have the following theorem.

Theorem 2.2. There is no nonzero parallel -form on an -Kenmotsu manifold.

Proof. We omit it.

A plane section in , , of a Kenmotsu manifold ( ) is called a -section if it spanned by a vector orthogonal to and . A connected Kenmotsu manifold is called a Kenmotsu space form and it is denoted by if it has the constant -sectional curvature . The curvature tensor of a Kenmotsu space form is given by for any .

Now, let be a -dimensional almost Hermitian manifold. A surjective map is called a contact-complex Riemannian submersion if it is a Riemannian submersion and satisfies [10]

In [13], we have proven the following theorem.

Theorem 2.3. Let be a contact-complex Riemannian submersion from a -dimensional Kenmotsu manifold to a -dimensional almost Hermitian manifold . Then, is a Kählerian manifold. Moreover, is a Kenmotsu space form if and only if is a complex space form.

3. Harmonic Maps and Stability

Let and be two Riemannian manifolds and a differentiable map. Then, the second fundamental form of is defined by where is the Levi-Civita connection on and is the connection induced by on the bundle , which is the pull-back of the Levi-Civita connection on , and satisfies the following formula (see [8]):

The tension field of is defined as the trace of the second fundamental form , that is , where is an orthonormal basis for at .

In what follows, we will use Einstein summation convention, so we will omit the sigma symbol.

We say that a map is a harmonic map .

Examples. (1) If is the circle , a map is harmonic if and only if it is a geodesic parametrized proportionally to arc length. (2) If , a harmonic map is a harmonic function. (3) A holomorphic map between two Kähler manifolds is harmonic [8]. For examples in the contact metric geometry, see [5, 6, 14].

Now let us consider a variation , with and . If the corresponding variation vector fields are denoted by and , the Hessian of is given by where is the canonical measure associated to the Riemannian metric and is a second-order self-adjoint operator acting on by where is the curvature operator on .

We say that a map from an almost contact metric manifold to an almost Hermitian manifold is a -holomorphic map if and only if .

If is a Sasaki manifold and is a Kähler manifold, then any -holomorphic map from to is a harmonic map [14].

Then, we can prove the same result for any -holomorphic map from an -Kenmotsu manifold to a Kähler manifold (see also [15]).

Our main result is the following.

Theorem 3.1. Let be a compact -Kenmotsu manifold and let be a Kähler manifold. Then, any -holomorphic map is stable.

If is compact, the spectrum of consists only of a discrete set of an infinite number of eigenvalues with finite multiplicities, bounded below by the first one. We define the Morse index of the harmonic map as the sum of multiplicities of negative eigenvalues of the Jacobi operator . Equivalently, the Morse index of equals the dimension of the largest subspace of on which the Hessian is negative definite (see [8, 9]).

We recall the following formula (see [5, 9]): where we omitted the summation symbol for repeated indices , [5].

Now, let be a local orthonormal -basis on such that .

From the -holomorphicity of and by , we have . Thus, from (3.5), we obtain the following.

Lemma 3.2. Let be a -holomorphic map from an -Kenmotsu manifold to a Kähler manifold . Then, one has

Lemma 3.3. Let be a vector field on such that for any , where and . Then,

Proof. Let By using (3.7) and (2.3) we obtain and (3.8) follows.

Proposition 3.4. Let be a compact -Kenmotsu manifold. Then, the function satisfies

Proof. We have Using (2.1)–(2.4), we obtain . Since is a compact manifold (without boundary), using Stokes's theorem, we have so that (3.11) follows from (3.13).

Now we are ready to prove Theorem  3.1 . Since is a -holomorphic map, by using the curvature Kähler identity on and Bianchi's identity, we have For any , we define the operator by the formula for any (see [5]).

Using Lemmas 3.2, 3.3, and (3.14), by a straightforward calculation, we obtain because

Thus, we have for any , so that is a stable harmonic map.

Acknowledgment

This work was partially supported by F.I.L., Parma University.

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