Harmonic Maps and Stability on 𝑓-Kenmotsu Manifolds

Abstract

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

1. Introduction

In Section 2, 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.

A 1-Kenmotsu manifold is a Kenmotsu manifold (see Kenmotsu [3, 4]. Theorem 2.1 provides a geometric interpretation of an 𝑓-Kenmotsu structure.

In Section 3, 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)−Tr_𝑔𝛽 holds, where 𝜏(𝐹) is the tension field of 𝐹 and 𝛽(𝑋,𝑌)=(∇^{0𝑥002𝑑𝑐}_𝑋𝐽)(𝐹_⋆𝑌), ^0𝑥0∇^02𝑑𝑐 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 Tr_𝑔𝛽=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 𝐽_𝐹, called the Jacobi operator, is associated to the harmonic map 𝐹. It is well known that the spectrum of 𝐽_𝐹 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 𝐽_𝐹 [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 𝜑²=−𝐼+𝜂⊗𝜉,𝜂(𝜉)=1,(2.1) where 𝐼 denotes the identity transformation.

It seems natural to include also 𝜑𝜉=0 and 𝜂◦𝜑=0; both can be derived from (2.1).

Let 𝑔 be an associated Riemannian metric on 𝑀 such that 𝑔(𝑋,𝑌)=𝑔(𝜑𝑋,𝜑𝑌)+𝜂(𝑋)𝜂(𝑌).(2.2) 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 by 𝐶^∞(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 that 𝑀 is an 𝑓-Kenmotsu manifold if there exists an almost contact metric structure (𝜑,𝜉,𝜂,𝑔) on 𝑀 satisfying (∇^{0𝑥002𝑑𝑐}_𝑋𝜑)𝑌=𝑓(𝑔(𝜑𝑋,𝑌)𝜉−𝜂(𝑌)𝜑𝑋)(2.3) for 𝑋,𝑌∈Γ(𝑇𝑀), where 𝑓 is a smooth function on 𝑀 such that 𝑑𝑓⋀𝜂=0.

A 1-Kenmotsu manifold is a Kenmotsu manifold [2, 3].

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

Theorem 2.1 (Olszak-Roşca). Let 𝑀 be an almost contact metric manifold. Then, 𝑀 is 𝑓-Kenmotsu if and only if it satisfies the following conditions:
(a) the distribution 𝐷=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 𝐿_𝜉𝜑=0.

Moreover, we have ^0𝑥0∇^02𝑑𝑐_𝑋𝜉=𝑓(𝑋−𝜂(𝑋)𝜉)(2.4) which gives div𝜉=2𝑛𝑓.

The characteristic vector field of an 𝑓-Kenmotsu manifold also satisfies 𝑅(𝑋,𝑌)𝜉=𝑓²(𝜂(𝑋)𝑌−𝜂(𝑌)𝑋).(2.5)

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 parallel 2-form on an 𝑓-Kenmotsu manifold.

Proof. We omit it.

A plane section 𝑝 in 𝑇𝑀_𝑥^{0𝑥002𝑑𝑐}, 𝑥∈𝑀^{0𝑥002𝑑𝑐}, 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 ^0𝑥𝑀^002𝑑𝑐 is called a Kenmotsu space form and it is denoted by ^0𝑥𝑀^002𝑑𝑐(𝑐) if it has the constant 𝜑-sectional curvature 𝑐. The curvature tensor of a Kenmotsu space form ^0𝑥𝑀^002𝑑𝑐(𝑐) is given by 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}(2.6) for any 𝑋,𝑌,𝑍∈Γ(𝑇𝑀^{0𝑥002𝑑𝑐}).

Now, let 𝑀(𝐽,𝑔^′) be a 2𝑚-dimensional almost Hermitian manifold. A surjective map 𝜋∶𝑀^{0𝑥002𝑑𝑐}→𝑀 is called a contact-complex Riemannian submersion if it is a Riemannian submersion and satisfies [10] 𝜋_⋆◦𝜑=𝐽◦𝜋_⋆.(2.7)

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

Theorem 2.3. Let 𝜋∶𝑀^{0𝑥002𝑑𝑐}→𝑀 be a contact-complex Riemannian submersion from a (2𝑚+1)-dimensional Kenmotsu manifold ^0𝑥𝑀^002𝑑𝑐 to a 2𝑚-dimensional almost Hermitian manifold 𝑀. Then, 𝑀 is a Kählerian manifold. Moreover, ^0𝑥𝑀^002𝑑𝑐 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 𝛼_𝐹 of 𝐹 is defined by 𝛼_𝐹(𝑋,𝑌)=∇^{0𝑥002𝑑𝑐}_𝑋𝐹_⋆𝑌−𝐹_⋆(∇_𝑋𝑌),(3.1) where ∇ is the Levi-Civita connection on 𝑀 and ^0𝑥0∇^02𝑑𝑐 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]): ^0𝑥0∇^02𝑑𝑐_𝑋𝐹_⋆𝑌−∇^{0𝑥002𝑑𝑐}_𝑌𝐹_⋆𝑋=𝐹_⋆([𝑋,𝑌]),𝑋,𝑌∈Γ(𝑇𝑀).(3.2)

The tension field 𝜏(𝐹) of 𝐹 is defined as the trace of the second fundamental form 𝛼_𝐹, that is 𝜏(𝐹)_𝑥=∑𝛼_𝐹(𝑒_𝑖,𝑒_𝑖)(𝑥), where (𝑒₁,…,𝑒_𝑚) is an orthonormal basis for 𝑇_𝑥𝑀 at 𝑥∈𝑀.

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

We say that a map 𝐹∶𝑀→𝑁 is a harmonic map 𝜏(𝐹) 𝑥∈𝑀.

Examples. (1) If 𝑀 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 variation 𝐹_𝑠,𝑡∈𝐶^∞(𝑀,𝑁), with 𝑠,𝑡∈(−𝜀,𝜀) and 𝐹_0,0=𝐹. If the corresponding variation vector fields are denoted by 𝑉 and 𝑊, the Hessian of 𝐹 is given by 𝐻_𝐹(𝑉,𝑊)=_𝑀ℎ(𝐽_𝐹(𝑉),𝑊)𝒱_𝑔,(3.3) where 𝒱_𝑔 is the canonical measure associated to the Riemannian metric 𝑔 and 𝐽_𝐹(𝑉) is a second-order self-adjoint operator acting on Γ(𝐹⁻¹(𝑇𝑁)) by 𝐽_𝐹(𝑉)=_𝑖(∇^{0𝑥002𝑑𝑐}_∇𝑒𝑖𝑒_𝑖−∇^{0𝑥002𝑑𝑐}∇_𝑒𝑖^{0𝑥002𝑑𝑐}_𝑒𝑖)𝑉−_𝑖𝑅^′(𝑉,𝐹_⋆𝑒_𝑖)𝐹_⋆𝑒_𝑖,(3.4) where 𝑅^′ is the curvature operator on (𝑁,ℎ).

We say that a map 𝑓∶(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_⋆.

If 𝑀(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. Let 𝑀(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.

If 𝑀 is compact, the spectrum of 𝐽_𝐹 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 operator𝐽_𝐹. Equivalently, the Morse index of 𝐹 equals the dimension of the largest subspace of Γ(𝑓⁻¹(TN)) on which the Hessian 𝐻_𝐹 is negative definite (see [8, 9]).

We recall the following formula (see [5, 9]): 𝐻_𝐹(𝑉,𝑊)=_𝑀(ℎ(∇^{0𝑥002𝑑𝑐}_𝑒𝑎𝑉,∇^{0𝑥002𝑑𝑐}_𝑒𝑎𝑊)+ℎ(𝑅^′(𝐹_⋆𝑒_𝑎,𝑉)𝐹_∗𝑒_𝑎,𝑊))𝒱_𝑔,(3.5) where we omitted the summation symbol for repeated indices 𝑎=1,…,𝑛, 𝑛=dim𝑀 [5].

Now, let (𝑒₁,…,𝑒_𝑚;𝑓₁,…,𝑓_𝑚,𝜉) be a local orthonormal 𝜑-basis on 𝑀(𝜑,𝜉,𝜂,𝑔) such that 𝑓_𝑖=𝜑𝑒_𝑖,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. Let 𝐹∶𝑀→𝑁 be a (𝜑,𝐽)-holomorphic map from an 𝑓-Kenmotsu manifold 𝑀 to a Kähler manifold 𝑁. Then, one has 𝐻_𝐹=(𝑉,𝑉)_𝑀(ℎ(∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝑉,∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝑉)+ℎ(∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉,∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉))𝒱_𝑔+_𝑀(ℎ(𝑅^′(𝐹_⋆𝑒_𝑖,𝑉)𝐹_⋆𝑒_𝑖,𝑉)+ℎ(𝑅^′(𝐹_⋆𝑓_𝑖,𝑉)𝐹_⋆𝑓_𝑖,𝑉))𝒱_𝑔.(3.6)

Lemma 3.3. Let 𝑇 be a vector field on 𝑀 such that 𝑔(𝑇,𝑋)=ℎ(∇^{0𝑥002𝑑𝑐}_𝜑𝑋𝑉,𝐽𝑉)(3.7) for any 𝑋∈Γ(𝐷), where 𝐷=Ker𝜂 and 𝑔(𝑇,𝜉)=0. Then, div(𝑇)=ℎ(𝑅^′(𝐹_⋆𝑒_𝑖,𝐹_⋆𝑓_𝑖)𝑉,𝐽𝑉)+2ℎ(∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝐽𝑉,∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉).(3.8)

Proof. Let ℎ(𝑅^′(𝐹_⋆𝑒_𝑖,𝐹_⋆𝑓_𝑖)𝑉,𝐽𝑉)=ℎ(∇^{0𝑥002𝑑𝑐}∇_𝑒𝑖^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉−∇^{0𝑥002𝑑𝑐}∇_𝑓𝑖^{0𝑥002𝑑𝑐}_𝑒𝑖𝑉−∇^{0𝑥002𝑑𝑐}_{[𝑒𝑖,𝑓𝑖]}𝑉,𝐽𝑉)=𝑒_𝑖ℎ(∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉,𝐽𝑉)−ℎ(∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉,∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝐽𝑉)−𝑓_𝑖ℎ(∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝑉,𝐽𝑉)+ℎ(∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝑉,∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝐽𝑉)−ℎ(∇^{0𝑥002𝑑𝑐}_{∇𝑒𝑖𝑓𝑖}𝑉,𝐽𝑉)+ℎ(∇^{0𝑥002𝑑𝑐}_{∇𝑓𝑖𝑒𝑖}𝑉,𝐽𝑉).(3.9) By using (3.7) and (2.3) we obtain div(𝑇)=𝑔(∇_𝑒𝑖𝑇,𝑒_𝑖)+𝑔(∇_𝑓𝑖𝑇,𝑓_𝑖)+𝑔(∇_𝜉𝑇,𝜉)=𝑒_𝑖𝑔(𝑇,𝑒_𝑖)−𝑔(𝑇,∇_𝑒𝑖𝑒_𝑖)+𝑓_𝑖𝑔(𝑇,𝑓_𝑖)−𝑔(𝑇,∇_𝑓𝑖𝑓_𝑖)=𝑒_𝑖ℎ(∇^{0𝑥002𝑑𝑐}_𝑓𝑖𝑉,𝐽𝑉)−𝑓_𝑖ℎ(∇^{0𝑥002𝑑𝑐}_𝑒𝑖𝑉,𝐽𝑉)+ℎ(∇^{0𝑥002𝑑𝑐}_{∇𝑓𝑖𝑒𝑖}𝑉,𝐽𝑉)+ℎ(∇^{0𝑥002𝑑𝑐}_{∇𝑒𝑖𝑓𝑖}𝑉,𝐽𝑉)(3.10) and (3.8) follows.

Proposition 3.4. Let 𝑀(𝜑,𝜉,𝜂,𝑔) be a compact 𝑓-Kenmotsu manifold. Then, the function 𝑓 satisfies _𝑀𝑓𝒱_𝑔=0.(3.11)

Proof. We have div(𝜉)=𝑔(𝑒_𝑖,∇_𝑒𝑖𝜉)+𝑔(𝑓_𝑖,∇_𝑓𝑖𝜉)+𝑔(𝜉,∇_𝜉𝜉).(3.12) Using (2.1)–(2.4), we obtain div(𝜉)=−2𝑛𝑓. Since 𝑀 is a compact manifold (without boundary), using Stokes's theorem, we have _𝑀div(𝜉)𝒱_𝑔=0,(3.13) so that (3.11) follows from (3.13).

Now we are ready to prove Theorem  3.1 . 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 𝑅^′(𝐹_⋆𝑒_𝑖,𝑉)𝐹_⋆𝑒_𝑖+𝑅^′(𝐹_⋆𝑓_𝑖,𝑉)𝐹_⋆𝑓_𝑖=−𝐽𝑅^′(𝐹_⋆𝑒_𝑖,𝐹_⋆𝑓_𝑖𝑉).(3.14) For any 𝑉∈Γ(𝐹⁻¹(𝑇𝑁)), we define the operator 𝐷𝑉∶Γ(𝑇𝑀)→Γ(𝐹⁻¹(𝑇𝑁))(3.15) by the formula 𝐷𝑉(𝑋)=∇^{0𝑥002𝑑𝑐}_𝜑𝑋𝑉−𝐽∇^{0𝑥002𝑑𝑐}_𝑋𝑉,(3.16) for any 𝑋∈Γ(𝑇𝑀) (see [5]).

Using Lemmas 3.2, 3.3, and (3.14), by a straightforward calculation, we obtain 𝐻_𝐹1(𝑉,𝑉)=2_𝑀(ℎ(𝐷𝑉(𝑒_𝑖),𝐷𝑉(𝑒_𝑖))+ℎ(𝐷𝑉(𝑓_𝑖),𝐷𝑉(𝑓_𝑖)))𝒱_𝑔(3.17) because ∫_𝑀div(𝑇)𝒱_𝑔=0.

Thus, we have 𝐻_𝐹(𝑉,𝑉)≥0 for any 𝑉∈Γ(𝐹⁻¹(𝑇𝑁)), so that 𝐹 is a stable harmonic map.

Acknowledgment

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