International Journal of Mathematics and Mathematical Sciences

Volume 2008 (2008), Article ID 798317, 7 pages

http://dx.doi.org/10.1155/2008/798317

## 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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