Harmonic Maps and Bi-Harmonic Maps on CR-Manifolds and Foliated Riemannian Manifolds ()
1. Introduction
The theory of harmonic maps has been extensively developed and applied in many problems in topology and differential geometry (cf. [1] [2] [3] , etc.). Eells and Lemaire raised ( [3] ) a problem to study
-harmonic maps and G. Y. Jiang calculated [4] the first variational and second formulas of the bienergy.
On the other hand, B.Y. Chen proposed [5] the famous conjecture in the study of sub-manifolds in the Euclidean space. B. Y. Chen’s conjecture and the generalized B. Y. Chen’s conjecture are as follows:
The B. Y. Chen’s conjecture: Every biharmonic isometric immersion into the Eucli- dean space
must be harmonic (minimal).
The generalized B. Y. Chen’s conjecture: Every biharmonic isometric immersion of a Riemannian manifold
into a Riemannian manifold
of non-positive curvature must be harmonic (minimal).
The B. Y. Chen’s conjecture is still open, but the generalized B. Y. Chen’s conjecture was solved negatively by Ye-Lin Ou and Liang Tang [6] , due to several authors have contributed to give partial answers to solve these problems (cf. [7] - [17] ).
For the first and second variational formula of the bienergy, see [4] .
Then, the CR analogue for harmonic maps and biharmonic maps has been raised as follows.
The CR analogue of the generalized Chen’s conjecture: Let
be a complete strictly pseudoconvex CR manifold, and
, a Riemannian manifold of non-positive curvature. Then, every pseudo biharmonic isometric immersion
must be pseudo harmonic.
For the works on CR analogue of biharmonic maps, see [18] [19] [20] . We will show (cf. [20] ):
Theorem 1.1. (cf. Theorem 2.1) Let
be a pseudo biharmonic map of a strictly pseudoconvex complete CR manifold
into another Riemannian manifold
of non positive curvature.
If
has finite pseudo bienergy
and finite pseudo energy
, then it is pseudo harmonic, i.e.,
.
Next, let us consider the analogue of harmonic maps and biharmonic maps for foliations are also given as follows. Transversally biharmonic maps between two foliated Riemannian manifolds were introduced by Chiang and Wolak (cf. [21] ) and see also [22] [23] [24] [25] [26] . They are generalizations of transversally harmonic maps introduced by Konderak and Wolak (cf. [27] [28] ).
Among smooth foliated maps
between two Riemannian foliated manifolds, one can define the transversal energy and derive the Euler-Lagrange equation, and transversally harmonic map as its critical points which are by definition the transversal tension field vanishes,
. The transverse bienergy can be also defined as
whose Euler-Lagrange equation is that the transversal biten-
sion field
vanishes and the transversally biharmonic maps which are, by definition, vanishing of the transverse bitension field.
Recently, S.D. Jung studied extensively the transversally harmonic maps and the transversally biharmonic maps on compact Riemannian foliated manifolds (cf. [29] [30] [31] [32] ).
Then, we will study transversally biharmonic maps of a complete (possibly non- compact) Riemannian foliated manifold
into another Riemannian foliated manifold
of which transversal sectional curvature is non-positive. Then, we will show (cf. [33] ) that:
Theorem 1.2. (cf. Theorem 2.6) Let
and
be two Riemannian foliated manifolds, and assume that the transversal sectional curvature of
is non-positive. Let
be a smooth foliated map which is an isometric immersion of
into
. If
is transversally biharmonic with the finite transversal energy
and finite transversal bienergy
, then it is transversally harmonic.
Finally, in Section 5, instead of isometric immersions, we will consider a principal G-bundle
, and show a new result whose details will be appeared in [34] .
Theorem 1.3. (cf. Theorem 5.1) Let
be a principal G-bundle over a Riemannian manifold
whose Ricci tensor is negative definite. Then, if
is biharmonic, then it is harmonic.
2. Preliminaries
2.1. First and Second Variational Formulas for the Energy
First, let us recall the theory of harmonic maps. For a smooth map
of a Riemannian manifold
into another Riemannian manifold
, the energy functional
is defined by
(2.1)
whose first variational formula is:
(2.2)
Here, V is a variational vector field is given by
,
,
and the tension field
is given by
(2.3)
(2.4)
where
and
are Levi-Civita connections of
and
, respectively. Then,
is harmonic if
.
The second variation formula of the energy functional
for a harmonic map
is:
(2.5)
where
(2.6)
(2.7)
where
is a locally defined frame field on
. The
-energy functional due to J. Eells and L. Lemaire ( [1] [2] [3] ) is
(2.8)
which turn out that
(2.9)
Furthermore, the first variation formula for
is (cf. [4] ):
(2.10)
(2.11)
Then, one can define that
is biharmonic (cf. [4] ) if
.
2.2. The CR Analogue of the Generalized Chen’s Conjecture
In this part, we first raise the CR analogue of the generalized Chen’s conjecture, and settle it for pseudo biharmonic maps with finite pseudo energy and finite pseudo bienergy.
Let us recall a strictly pseudoconvex CR manifold (possibly non compact)
of
-dimension, and the Webster Riemannian metric
given by
![]()
for
. Recall the material on the Levi-Civita connection
of
. Due to Lemma 1.3, Page 38 in [35] , it holds that,
(2.12)
where
is the Tanaka-Webster connection,
, and
,
, and
is the torsion tensor of
. And also,
,
for all vector fields X, Y on M. Here, J is the
complex structure on
and is extended as an endomorphism on
by
.
Then, we have
(2.13)
(2.14)
where
is a locally defined orthonormal frame field on
with respect to gθ, and T is the characteristic vector field of
. For (3.6), it follows from that
,
and
since
. For (3.7), notice that the Tanaka-Webster connection
satisfies
, and also
and JT = 0, so that
which imply (3.7).
Let us consider the generalized B.-Y. Chen’s conjecture for pseudo biharmonic maps which is CR analogue of the usual generalized Chen’s conjecture for biharmonic maps:
The CR analogue of the generalized B.-Y. Chen’s conjecture for pseudo bihar- monic maps:
Let
be a complete strictly pseudoconvex CR manifold, and assume that
is a Riemannian manifold of non-positive curvature.
Then, every pseudo biharmonic isometric immersion
must be pseudo harmonic.
Then, we will show:
Theorem 2.1. Assume that φ is a pseudo biharmonic map of a strictly pseudoconvex complete CR manifold
into another Riemannian manifold
of non positive curvature.
If φ has finite pseudo bienergy
and finite pseudo energy
, then it is pseudo harmonic, i.e.,
.
2.3. The Green’s Formula on a Foliated Riemannian Manifold
Then, we prepare the materials for the first and second variational formulas for the transversal energy of a smooth foliated map between two foliated Riemannian manifolds following [31] [32] [36] . Let
be an
-dimensional foliated Riemannian manifold with foliation
of codimension q and a bundle-like Riemannian metric g with respect to
(cf. [37] [38] ). Let TM be the tangent bundle of M, L, the tangent bundle of
, and Q = TML, the corresponding normal bundle of
. We denote
the induced Riemannian metric on the normal bundle Q, and
, the transversal Levi-Civita connection on Q,
, the transversal curvature tensor, and
, the transversal sectional curvature, respectively. Notice that the bundle projection
is an element of the space
of Q-valued 1-forms on M. Then, one can obtain the Q-valued bilinear form
on M, called the second fundamental form of
, defined by
![]()
The trace
of
, called the tension field of
is defined by
![]()
where
spanns
on a neighborhood U on M. The Green’s theorem, due to Yorozu and Tanemura [36] , of a foliated Riemannian manifold
says that
(2.15)
where
denotes the transversal divergence of
with respect to
given by
. Here
spanns
where
is the orthogonal complement bundle of L with a natural identification
.
2.4. The Variational Formulas for Foliations
Let
, and
be two compact foliated Riemannian manifolds. The transversal energy
among the totality of smooth foliated maps from
into
by
(2.16)
Here, a smooth map
is a foliated map is, by definition, for every leaf L of
, there exists a leaf
of
satisfying
. Then,
can be regarded as a section of
where
is a subspace of the cotangent bundle T*M. Here, π,
are the projections of
and
. Notice that our definition of the transversal energy is a slightly different from the one of Jung’s definition (cf. [32] , p. 5).
The first variational formula is given (cf. [?]), for every smooth foliated variation
with
and
in which
being a section
,
(2.17)
Here,
is the transversal tension field defined by
(2.18)
where
is the induced connection in
from the Levi-Civita connection of
, and
is a locally defined orthonormal frame field on Q.
Definition 2.2. A smooth foliated map
is said to be transversally harmonic if
.
Then, for a transversally harmonic map
, the second variation formula of the transversal energy
is given as follows (cf. [?], p. 7) : let
be any two parameter smooth foliated variation of ![]()
with
,
and
,
(2.19)
where
is a second order semi-elliptic differential operator acting on the space
of sections of
which is of the form:
(2.20)
for
. Here,
is the Levi-Civita connection of
, and recall also that:
(2.21)
(2.22)
Definition 2.3. The transversal bitension field
of a smooth foliated map
is defined by
(2.23)
Definition 2.4. The transversal bienergy E2 of a smooth foliated map
is defined by
(2.24)
Remark that this definition of the transversal bienergy is also slightly different from the one of Jung (cf. Jung [32] , p. 13, Definition 6.1). On the first variation formula of the transversal bienergy is given as follows. For a smooth foliated map φ and a smooth foliated variation
of
, it holds (cf. [32] , p. 13) that
(2.25)
Definition 2.5. A smooth foliated map
is said to be transversally biharmonic if
.
Then, one can ask the following generalized B.Y. Chen’s conjecture:
The generalized Chen’s conjecture:
Let
be a transversally biharmonic map from a foliated Riemannian manifold
into another foliated Riemannian manifold
whose transversal sectional curvature
is non-positive. Then,
must be transversally harmonic.
Then, we can state our main theorem which gives an affirmative partial answer to the above generalized Chen’s conjecture under the additional assumption that
has both the finite transversal energy and the finite transversal bienergy:
Theorem 2.6. Let
a smooth foliated map which is an isometric immersion of
into
. Assume that
is complete (possibly non-compact), and the transversal sectional curvature
of
is non-positive.
If φ is transversally biharmonic having both the finite transversal energy
and the finite transversal bienergy
, then it is transversally harmonic.
Remark that in the case that
is compact, Theorem 2.5 is true due to Jung’s work (cf. [32] Theorem 6.4, p. 14).
3. Proof of Theorem 2.1
The proof of Theorem 2.1 is divided into several steps which will appear in [20] .
(The first step) For an arbitrarily fixed point
, let
where
is a distance function on
, and let us take a cut off function
on
, i.e.,
(3.1)
where r is the distance function from
, and
is the Levi-Civita connection of
, respectively. Assume that
is a pseudo biharmonic map, i.e.,
(3.2)
(The second step) Then, we have
(3.3)
In (3.3), notice that
is the sectional curvature of
corresponding to the vectors
and
. Since
has the non-positive sectional curvature, (3.3) is non-positive.
On the other hand, for the left hand side of (3.3), it holds that
(3.4)
Here, let us recall, for
,
![]()
where
is a locally defined orthonormal frame field of
and
is defined by
![]()
for
and
. Here,
is the
-component of
corresponding to the decomposition of
, and
is the induced connection of
from the Levi-Civita con- nection
of
.
Since
(3.5)
the right hand side of (3.4) is equal to
(3.6)
Therefore, together with (3.3), we have
(3.7)
where we define
by
![]()
Then, it holds that
for every
which implies that
![]()
Therefore, we have that
The right hand side of (3.7)
(3.8)
foe every
. By taking
, we obtain
(3.9)
Therefore, we obtain, due to the properties that
on
, and
,
(3.10)
(The third step) By our assumption that
and
is complete, if we let
, then
goes to M, and the right hand side of (3.10) goes to zero. We have
(3.11)
This implies that
(3.12)
(The fourth step) Let us take a 1 form
on M defined by
![]()
Then, we have
(3.13)
where we put
,
![]()
and
(3.14)
Furthermore, let us define a
function
on M by
(3.15)
where
is the Tanaka-Webster connection. Notice that
(3.16)
where
is the natural projection. We used the facts that
, and
( [35] , p.37). Here, recall again
is the Levi-Civita connection of
, and
is the Tanaka-Webster connection. Then, we have, for (3.16),
(3.17)
We used (3.12)
to derive the last second equality of (3.17). Then, due to (3.17), we have for
,
(3.18)
In the last equality, we used Gaffney’s theorem ( [16] , p. 271, [?]).
Therefore, we obtain
, i.e.,
is pseudo harmonic.
We obtain Theorem 2.1.
4. Proof of Main Theorem 2.6
In this section, we give a proof of Theorem 2.6 which will appear in [34] , by a similar way to the case of foliations as Theorem 2.1.
(The first step) First, let us take a cut off function
from a fixed point
on
, i.e.,
![]()
where
,
is a distance function from
on
,
is the Levi-Civita connection of
, respectively.
Assume that
is a transversally biharmonic map of
into
, i.e.,
(4.1)
where recall
is the induced connection on
.
(The second step) Then, by (4.1), we obtain that
(4.2)
where the sectional curvature
of
corresponding to the plane spanned by
and
is non-positive.
(The third step) On the other hand, the left hand side of (4.2) is equal to
(4.3)
since
![]()
Together (4.2) and (4.3), we obtain
(4.4)
Because, putting
, ![]()
, we have
![]()
which is
(4.5)
If we put
in (4.5), then we obtain
(4.6)
By (4.6), we have the second inequality of (4.4).
(The fourth step) Noticing that η = 1 on
and
in the inequality
(3.4), we obtain
(4.7)
Letting
, the right hand side of (4.7) converges to zero since
. But due to (4.7), the left hand side of (4.7) must converge to
since
tends to M because
is complete.
Therefore, we obtain that
![]()
which implies that
(4.8)
(The fifth step) Let us define a 1-form
on M by
(4.9)
and a canonical dual vector field
on M by
. Then, its divergence
written as
,
can be given as follows. Here,
and
are locally defined orthonormal frame fields on leaves L of
and Q, respectively, (
,
,
).
Then, we can calculate
as follows:
(4.10)
since
in the last equality of (4.10). Integrating the both hands of (4.10) over M, we have
(4.11)
because of
. Notice that both hands in (4.11) are well defined because of
and
.
Since
is the second fundamental form of each leaf L in
and
(4.12)
the right hand side of (4.11) coincides with
(4.13)
(4.11) is equivalent to that
(4.14)
If
is an isometric immersion, then it holds that
, which implies that both the left hand side and the second term of the right hand side of (4.14) vanish, that is,
. Therefore
.
We obtain Theorem 2.6.
5. Principal G-Bundles
In this section, we show the following theorem which is quite new and the more detail [34] will appear elsewhere.
Theorem 5.1 Let
be a principal G-bundle over a Riemannian manifold
whose Ricci tensor is negative definite. Then, if
is biharmonic, then it is harmonic.
Let us consider a principal G-bundle
whose the total space P is compact. Assume that the projection
is biharmonic, which is by definition,
, where
is the tension field of
which is defined by
(5.1)
the Jacobi operator J is defined by
(5.2)
is the rough Laplacian defined by
(5.3)
and
(5.4)
where
is a locally defined orthonormal frame field on
.
The tangent space
is canonically decomposed into the orthogonal direct sum of the vertical subspace
and the horizontal subspace
:
. Then, we have
![]()
where
, respectively. Then, we obtain
![]()
Therefore, we obtain
(5.5)
where we denote by
, the sectional curvature of
through two plane of
given by
, and
is the Ricci curvature of
along
. The left hand side of (5.5) is non-negative, and then, the both hand sides of (5.5) must vanish if the Ricci curvature of
is non-positive. Therefore, we obtain
(5.6)
Let us consider a 1-form
on M defined by
(5.7)
Then, for every
, we have
(5.8)
which implies that
is parallel 1-form on
. Since we assume that the Ricci tensor of
is negative definite,
must vanish (so called Bochner’s theorem, cf. [40] , p. 55). Therefore,
, i.e., the projection of the principal G-bundle
is harmonic. We obtain Theorem 5.1.
Acknowledgements
None.
Funding
Supported by the Grant-in-Aid for the Scientific Research, (C) No. 25400154, Japan Society for the Promotion of Science.