1. Introduction
Harmonic maps between Riemannian manifolds were first introduced by Eells and Sampson in 1964. They showed that any map
from any compact Riemannian manifold
into a Riemannian manifold
with non-positive sectional curvature can be deformed into a harmonic maps. This is so-called the fundamental existence theorem for harmonic maps. In view of physics, harmonic maps have been studied in various fields of physics, such as super conductor, ferromagnetic material, liquid crystal, etc. [
1,
2,
3,
4,
5,
6,
7,
8,
9].
Lichnerowicz first studied
f-harmonic maps between Riemannian manifolds as a generalization of harmonic maps in 1970 [
10]. Recently, in Cherif et al. [
11], the researchers proved that any stable
f-harmonic map
from sphere
to Riemannian manifold
N is constant. Course [
12] studied the
f-harmonic flow on surfaces. Ou [
13] analysed the f-harmonic morphisms as a subclass of harmonic maps which pull back harmonic functions to f-harmonic functions. Many scholars have studied and done research on
f-harmonic maps, see for instance, [
10,
11,
13,
14,
15,
16,
17].
The concept of harmonic maps from a Finsler manifold to a Riemannian manifold was first introduced by Mo [
18]. On the workshop of Finsler Geometry in 2000, Professor S. S. Chern conjectured that the fundamental existence theorem for harmonic maps on Finsler spaces is true. In [
19], Mo and Yang, the researchers have proved this conjecture and shown that any smooth map from a compact Finsler manifold to a compact Riemannian manifold of non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class. Shen and Zhang [
20] extended Mo’s work to Finsler target manifold and obtained the first and second variation formulas.
As an application, He and Shen [
21] proved that any harmonic map from an Einstein Riemannian manifold to a Finsler manifold with certain conditions is totally geodesic and there is no stable harmonic map from an Euclidean unit sphere
to any Finsler manifolds. Harmonic maps between Finsler manifolds have been studied extensively by various researchers, see for instance, [
18,
19,
20,
21,
22].
In [
23], J. Lie introduced the notion of
-harmonic maps between Finsler manifolds. Let
be a
function such that
on
. The smooth map
is said to be
-harmonic if it is an extermal point of the
-energy functional
where
is the energy density of
,
denotes the volume of the standard
-dimensional sphere and
is the canonical volume element of
. The
-energy functional is the energy , the p-energy, the
energy of Sacks-Uhlenbeck and exponential energy when
is equal to
t,
,
and
, respectively [
20,
21,
23,
24,
25].
In view of physics, when
is a Riemannian manifold,
p-harmonic maps have been extensively applied in image processing for denoising color images [
26,
27]. Furthermore, exponential harmonic maps have been studied on gravity [
28]. The concept of
-harmonic maps, as an extension of harmonic,
p-harmonic and exponential harmonic maps, have an important role in physics and physical cosmology. For instance, instead of the scalar field in the Lagrangian, some of the
-harmonic maps, such as the trigonometric functions, are studied in order to reproduce the inflation. Moreover, there are other
-harmonic maps, such as exponential harmonic maps, are investigated in order to depict the phenomenon of the quintessence [
5,
29,
30].
Let
be a smooth map from a Finsler manifold
into a Riemannian manifold
and
be a smooth positive function. A map
is said to be
-harmonic if it is a critical point of the
-energy functional
where
g is the fundamental tensor of
,
is the pull-back of the metric
h by the map
and
is a smooth function given by
. By considering the Euler-Lagrange equation associated to the
-energy functional, it can be seen that any
-harmonic map
from a Finsler manifold
into a Riemannian manifold
without critical points (i.e.,
for all
), is a
-harmonic map with
.
In particular, when
and
is a Riemannian manifold,
-harmonic maps can be considered as the stationary solutions of inhomogeneous Heisenberg spin system, see for instance [
13,
14]. Furthermore, the intersection of
-harmonicity with curvature conditions justifies their application for gleaning valuable information on weighted manifolds and gradient Ricci solitons, see [
15,
16,
17].
The current paper is organized as follows. In
Section 2, a few concepts of Finsler geometry are reviewed. In
Section 3, the
-energy functional of a smooth map from a Finsler manifold to a Riemannian manifold is introduced and the corresponding Euler-Lagrange equation is obtained via calculating the first variation formula of the
-energy functional and an example is given. In
Section 4, the second variation formula of the
-energy functional for a
-harmonic map is derived. As an application, the stability theorems for
-harmonic maps are given.
2. Preliminaries
Throughout this paper, let
be an
m-dimensional smooth, oriented, compact Finsler manifold without boundary. In the local coordinates
on
, the fundamental tensor of
is defined as follows:
In this paper, the following conventions of index ranges are used
Let
be the natural projection on the projective sphere bundle
. The Finsler structure
F determines two important quantities on the pull-back bundle
as follows:
which are called the Hilbert form and Cartan tensor, respectively. The dual of the Hilbert form
is the distinguished section
of the pull-back bundle
. Note that, all indices related to
are raised and lowered with the metric
g.
On the pull-back bundle
, there exists uniquely the Chern connection
whose connection 1-forms
are satisfied the following equations
where
[
31]. Here
and
are the formal Christoffel symbols of the second kind for
. The curvature 2-forms of the Chern connection,
, have the following structure
See [
22] for an expository proof. The Landsberg curvature of
is defined as follows:
By [
22], we have
where “.” denotes the covariant derivative along the Hilbert form. Consider a
g-orthonormal frame
for any fibre of
where
is the Hilbert form
, and let
be its dual frame where
is the distinguished section
ℓ dual to the Hilbert form
. Set
where
. It can be seen that
is a local basis for the cotangent bundle
and
is a local basis for
. By (
2), it can be seen that
Tangent vectors on
which are annihilated by all
’s form the horizontal sub-bundle
of
. The fibres of
are
m-dimensional. On the other hand, let
be the vertical sub-bundle of
, its fibres are
dimensional. The decomposition
holds because
and
are direct summands, (see [
32], p. 7). The inner product
on
induces a Riemannian metric
on
as follows:
Furthermore, the volume element of with respect to is defined as follows
Lemma 1. [18] For , we have
where “∣” denotes the horizontal covariant derivative with respect to the Chern connection and is defined in (2). 3. The First Variation Formula
Let be a smooth map from a Finsler manifold into a Riemannian manifold , a natural projection on and . In the sequel, we denote the Chern connection on by , the connection induced by the Chern connection of on the pulled-back bundle over by ∇ and the Levi-Civita connection on by .
Let
be a smooth positive function. The
λ-energy density of
is the function
, defined by
where
is the pull-back of
h by the map
and
stands for taking the trace with respect to
g (the fundamental tensor of
F) at
By making use of (
5), the
-energy functional is defined as follows:
where
denotes the volume of the standard
-dimensional sphere and
is the canonical volume element of
. A map
is said to be
λ-harmonic if it is a critical point of the
-energy functional.
Let
be a smooth variation of
with the variational vector field
For any
, in local coordinate
on
M and
on
N, the
-energy density of
can be written as follows:
where
and
. Due to the fact that
is independent of
y and using (
7), we have
where
is a smooth function given by
. By using the definition of the gradient operator, we get
where
and
is the smooth function
. The function
is called the
energy density of
. Let
, which is a section of
. By Lemma 4, we have
where
is a section of
. By (
10), we get
By substituting (
9) and (
12) in (
8) and considering the Green’s theorem, the first variation formula of the
-energy functional is obtained as follows:
where
Here
denotes the horizontal part of
and
K is defined by (
11). The field
is said to be the
λ-tension field of
.
Theorem 1. Let be a smooth map from a Finsler manifold to a Riemannian manifold and . Then, ψ is the λ-harmonic map if and only if .
Example 1. Assume that is a locally Minkowski manifold and be the three-dimensional Euclidean space. Consider the map defined by Let be a positive smooth map such that . Due to the fact that the Landsberg curvature of locally Minkowski manifold vanishes and considering Theorem 1 and Equation (13), one can see that ψ is λ-harmonic.
Now, we discuss the relation between -harmonic maps and -harmonic maps from a Finsler manifolds to a Riemannian manifolds. Let be a function such that on . The smooth map is said to be -harmonic if it is an extermal point of the -energy functional
The concept of
-harmonic maps from a Finsler manifold was first introduced by J. Li [
23] in 2010. The
-energy functional is the energy , the p-energy, the
-energy of Sacks-Uhlenbeck and exponential energy when
is equal to
t,
,
and
, respectively. The Euler-Lagrange equation associated to
-energy functional is given by
For more details, see [
23]. The field
is called the
-tension field of
.
Proposition 1. Let be an -harmonic map from a Finsler manifold to a Riemannian manifold without critical points (i.e., for all ). Then, ψ is a λ-harmonic map with .
Proof. It is obtained from (
13) and (
15) immediately. ☐