# Harmonic Maps and Stability onf-Kenmotsu Manifolds

Authors: Vittorio Mangione
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

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


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## Body

## 1. Introduction

In Section2, we give preliminaries on f-Kenmotsu manifolds. The concept of f-Kenmotsu manifold, where f 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 f-Kenmotsu
manifold by the formula (2.3), where
f is a function on
M such that df⋀η=0. Here, η is the dual 1-form corresponding to the
characteristic vector field ξ of an almost contact metric structure on M. The condition df⋀η=0 follows in fact from
(2.3) if dimM≥5. This does not hold in general if dimM=3.A1-Kenmotsu manifold is a
Kenmotsu manifold (see
Kenmotsu [3, 4]. Theorem 2.1 provides a geometric interpretation
of an f-Kenmotsu
structure.In Section3, we initiate a study of harmonic maps
when the domain is a compact f-Kenmotsu manifold and the target is a Kähler
manifold.Ianus and Pastore [5, 6] defined a
(φ,J)-holomorphic map
between an almost contact metric manifold M(φ,η,ξ,g) and an almost
Hermitian manifold N(J,h) as a smooth map
F:M→N such that the condition F⋆◦φ=J◦F⋆ is satisfied. Then, the formula J(τ(F))=F⋆(divφ)−Trgβ holds, where
τ(F) is the tension
field of F and β(X,Y)=(∇˜XJ)(F⋆Y), ∇˜ being the
connection induced in the pull-back bundle F⋆(TN) (see [7]). It is easy to see that in our assumptions divφ=0 and Trgβ=0 so that a (φ,J)-holomorphic map
between an f-Kenmotsu
manifold M and a Kähler
manifold N is a harmonic
map. If M is a compact
manifold, a second-order elliptic operator JF, called the Jacobi operator, is associated
to the harmonic map F. It is
well known that the spectrum of JF 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 F as the sum of multiplicities of
negative eigenvalues of the Jacobi operator JF  [8, 9]. A harmonic map
is called
stable if the Morse index is zero. We have proven
that any (φ,J)-holomorphic map
from a compact f-Kenmotsu
manifold to a Kähler manifold is a stable harmonic map (see
[10]).


## 2.f-Kenmotsu Manifolds

A differentiable(2n+1)-dimensional
manifold M is said to have a
(φ,ξ,η)-structure or an
almost contact structure if there exist a tensor field φ of type (1, 1), a vector field
ξ, and a 1-form η on M satisfying
(2.1)φ2=−I+η⊗ξ,η(ξ)=1,
where
I denotes the identity
transformation.It seems natural to include alsoφξ=0 and η◦φ=0; both can be derived from
(2.1).Letg be an associated
Riemannian metric on M such
that
(2.2)g(X,Y)=g(φX,φY)+η(X)η(Y).
Putting
Y=ξ in (2.2) and using
(2.1), we get
η(X)=g(X,ξ), for any vector
field X on M.In this paper, we denote byC∞(M) and Γ(E) the algebra of
smooth functions on M and
the C∞(M)-module of
smooth sections of a vector bundle E, respectively. All manifolds are assumed to be connected
and of class C∞.
Tensors fields, distribution, and so on are assumed to be of class
C∞ if not
stated otherwise.We say thatM is an f-Kenmotsu manifold if there exists an almost
contact metric structure (φ,ξ,η,g) on M satisfying
(2.3)(∇˜Xφ)Y=f(g(φX,Y)ξ−η(Y)φX)
for
X,Y∈Γ(TM), where
f is a smooth function on
M such that df⋀η=0.A1-Kenmotsu manifold is a Kenmotsu
manifold [2, 3].The
following theorem provides a geometric interpretation of anyf-Kenmotsu structure.Theorem 2.1 (Olszak-Roşca).
LetM
be an almost contact metric manifold. Then,
M
is  f-Kenmotsu if
and only if it satisfies the following conditions:
(a)
the distributionD=Kerη is integrable and any leaf of the foliation
ℱ corresponding to
D is a totally umbilical
hypersurface with constant mean curvature;(b)
the almost Hermitian structure(J,g) induced on an
arbitrary leaf is Kähler;(c)
∇ξξ=0 and Lξφ=0.Moreover, we have(2.4)∇˜Xξ=f(X−η(X)ξ)  which
gives divξ=2nf.The characteristic vector field of anf-Kenmotsu manifold also satisfies
(2.5)R(X,Y)ξ=f2(η(X)Y−η(Y)X).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 f-Kenmotsu manifold we have the following theorem.Theorem 2.2.
There is no nonzero parallel2-form on an  f-Kenmotsu manifold.Proof.
We omit it.A plane sectionp in TxM˜,
x∈M˜, of a Kenmotsu manifold (f=1) is called a φ-section if it spanned by a
vector X orthogonal to
ξ and φX. A connected Kenmotsu manifold M˜ is called a
Kenmotsu space form and it is denoted by M˜(c) if it has the
constant φ-sectional
curvature c. The curvature
tensor of a Kenmotsu space form M˜(c) is given by
(2.6)4R(X,Y)Z=(c−3){g(Y,Z)X−g(X,Z)Y}+(c+1){η(X)η(Z)Y+−η(Y)η(Z)X+η(Y)g(X,Z)ξ−η(X)g(Y,Z)η+g(X,φZ)φY−g(Y,φZ)φX+2g(X,φY)φZ}
for any X,Y,Z∈Γ(TM˜).Now, letM(J,g′) be a 2m-dimensional almost Hermitian manifold. A surjective map
π:M˜→M
is called a
contact-complex Riemannian submersion if it is a
Riemannian submersion and satisfies [10]
(2.7)π⋆◦φ=J◦π⋆.In [13], we have proven the following theorem.Theorem 2.3.
Letπ:M˜→M
be a contact-complex Riemannian submersion from a
(2m+1)-dimensional
Kenmotsu manifold  M˜
to a  2m-dimensional almost Hermitian manifold  M. Then,  M
is a Kählerian
manifold. Moreover,  M˜
is a Kenmotsu space form if and only if  M
is a complex space form.


## 3. Harmonic Maps and Stability

Let(M,g) and (N,h) be two
Riemannian manifolds and F:M→N a differentiable map. Then, the second
fundamental form αF of F is defined
by
(3.1)αF(X,Y)=∇˜XF⋆Y−F⋆(∇XY),
where
∇ is the Levi-Civita
connection on M and
∇˜ is the connection
induced by F on the bundle
F−1(TN), which is the
pull-back of the Levi-Civita connection ∇' on N, and satisfies the following formula (see [8]): (3.2)∇˜XF⋆Y−∇˜YF⋆X=F⋆([X,Y]),X,Y∈Γ(TM).The
tension fieldτ(F) of F is defined as the trace of the second
fundamental form αF, that is τ(F)x=∑αF(ei,ei)(x), where
(e1,…,em) is an
orthonormal basis for TxM at x∈M.In what follows, we will use
Einstein summation convention, so we will omit the sigma symbol.We say that a mapF:M→N is
a harmonic mapτ(F)x∈M.Examples.
(1) IfM is the
circle S1, a map
F:S1→(N,g) is harmonic if
and only if it is a geodesic parametrized proportionally to arc
length. (2) If N=ℝ, a harmonic map F:(M,g)→ℝ 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 variationFs,t∈C∞(M,N), with s,t∈(−ε,ε) and F0,0=F. If the
corresponding variation vector fields are denoted by V and W, the Hessian of F is given by
(3.3)HF(V,W)=∫Mh(JF(V),W)𝒱g,
where
𝒱g is the
canonical measure associated to the Riemannian metric g and JF(V) is a
second-order self-adjoint operator acting on Γ(F−1(TN)) by
(3.4)JF(V)=∑i(∇˜∇eiei−∇˜ei∇˜ei)V−∑iR′(V,F⋆ei)F⋆ei,
where
R′ is the curvature operator on (N,h).We say
that a mapf:(M,φ,ξ,η,g)→(N,J,h) from an almost
contact metric manifold to an almost
Hermitian manifold
is a  (φ,J)-holomorphic
map if and only ifF⋆◦φ=J◦F⋆.IfM(φ,ξ,η,g) is a Sasaki
manifold and N(J,h) is a Kähler
manifold, then any (φ,J)-holomorphic map
from M to N is a harmonic map [14].Then, we can prove the same result for any(φ,J)-holomorphic map
from an f-Kenmotsu manifold
to a Kähler manifold (see also [15]).Our main result is the
following.Theorem 3.1.
LetM(φ,ξ,η,g)
be a compact  f-Kenmotsu manifold and let  N(J,h)
be a Kähler
manifold. Then, any  (φ,J)-holomorphic
map  F:M→N
is stable.IfM is compact, the spectrum of JF 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 F:M→N as the
sum of multiplicities of negative eigenvalues of the Jacobi
operatorJF. Equivalently, the Morse index of F equals the dimension of the largest
subspace of Γ(f−1(TN)) on which the Hessian HF is negative definite (see [8, 9]).We
recall the following formula (see [5, 9]):
(3.5)HF(V,W)=∫M(h(∇˜eaV,∇˜eaW)+h(R′(F⋆ea,V)F*ea,W))𝒱g,
where we
omitted the summation symbol for repeated indices a=1,…,n, n=dimM  [5].Now, let(e1,…,em;f1,…,fm,ξ) be a local
orthonormal φ-basis
on M(φ,ξ,η,g) such that
fi=φei,i=1,…,m.From the(φ,J)-holomorphicity
of F and by φξ=0, we have F⋆ξ=0. Thus, from
(3.5), we obtain the
following.Lemma 3.2.
LetF:M→N
be a  (φ,J)-holomorphic map from an  f-Kenmotsu
manifold  M
to a Kähler
manifold  N. Then, one has
(3.6)HF(V,V)=∫M(h(∇˜eiV,∇˜eiV)+h(∇˜fiV,∇˜fiV))𝒱g+∫M(h(R′(F⋆ei,V)F⋆ei,V)+h(R′(F⋆fi,V)F⋆fi,V))𝒱g.Lemma 3.3.
LetT
be a vector field on  M
such that
(3.7)g(T,X)=h(∇˜φXV,JV)
for any
X∈Γ(D), where
D=Kerη and g(T,ξ)=0. Then,
(3.8)div(T)=h(R′(F⋆ei,F⋆fi)V,JV)+2h(∇˜eiJV,∇˜fiV).Proof.
Let(3.9)h(R′(F⋆ei,F⋆fi)V,JV)=h(∇˜ei∇˜fiV−∇˜fi∇˜eiV−∇˜[ei,fi]V,JV)=eih(∇˜fiV,JV)−h(∇˜fiV,∇˜eiJV)−fih(∇˜eiV,JV)+h(∇˜eiV,∇˜fiJV)−h(∇˜∇eifiV,JV)+h(∇˜∇fieiV,JV).
By using (3.7) and (2.3) we obtain
(3.10)div(T)=g(∇eiT,ei)+g(∇fiT,fi)+g(∇ξT,ξ)=eig(T,ei)−g(T,∇eiei)+fig(T,fi)−g(T,∇fifi)=eih(∇˜fiV,JV)−fih(∇˜eiV,JV)+h(∇˜∇fieiV,JV)+h(∇˜∇eifiV,JV)
and (3.8) follows.Proposition 3.4.
LetM(φ,ξ,η,g)
be a compact  f-Kenmotsu manifold. Then, the function  f
satisfies
(3.11)∫Mf𝒱g=0.Proof.
We have(3.12)div(ξ)=g(ei,∇eiξ)+g(fi,∇fiξ)+g(ξ,∇ξξ). Using (2.1)–(2.4), we obtain
div(ξ)=−2nf. Since M is a compact manifold (without boundary), using Stokes's
theorem, we have
(3.13)∫Mdiv(ξ)𝒱g=0,
so that (3.11) follows from (3.13).Now we are ready to
prove Theorem3.1
. Since F is a (φ,J)-holomorphic
map, by using the curvature Kähler identity R′(U,V)JW=JR′(U,V)W on N(J,h) and Bianchi's
identity, we have
(3.14)R′(F⋆ei,V)F⋆ei+R′(F⋆fi,V)F⋆fi=−JR′(F⋆ei,F⋆fiV).
For any V∈Γ(F−1(TN)), we define the operator
(3.15)DV:Γ(TM)→Γ(F−1(TN))
by the formula
(3.16)DV(X)=∇˜φXV−J∇˜XV,
for any
X∈Γ(TM) (see [5]).Using Lemmas3.2, 3.3, and (3.14), by a straightforward calculation, we obtain
(3.17)HF(V,V)=12∫M(h(DV(ei),DV(ei))+h(DV(fi),DV(fi)))𝒱g
because
∫Mdiv(T)𝒱g=0.Thus, we haveHF(V,V)≥0 for any V∈Γ(F−1(TN)), so that F is a stable harmonic map.



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