Journal of Applied Mathematics and Physics, 2016, 4, 2272-2289 http://www.scirp.org/journal/jamp

ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2016.412219 December 29, 2016

Harmonic Maps and Bi-Harmonic Maps on CR-Manifolds and Foliated Riemannian Manifolds

Shinji Ohno1, Takashi Sakai2, Hajime Urakawa3

1Osaka City University Advanced Mathematical Institute (OCAMI), Osaka, Japan 2Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Japan 3Institute for International Education, Global Learning Center, Tohoku University, Sendai, Japan

Abstract This is a survey on our recent works on bi-harmonic maps on CR-manifolds and fo-liated Riemannian manifolds, and also a research paper on bi-harmonic maps prin-cipal G-bundles. We will show, (1) for a complete strictly pseudoconvex CR manifold ( ),M gθ , every pseudo bi-harmonic isometric immersion ( ) ( ): , ,M g N hθϕ → into a Riemannian manifold of non-positive curvature, with finite energy and finite bi- energy, must be pseudo harmonic; (2) for a smooth foliated map of a complete, pos-sibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, then it is transversally harmonic; (3) we will claim that the similar result holds for prin-cipal G-bundle over a Riemannian manifold of negative Ricci curvature.

Keywords Foliation, Divergence Theorem, Transversally Harmonic, Transversally Biharmonic

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 k -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:

How to cite this paper: Ohno, S., Sakai, T. and Urakawa, H. (2016) Harmonic Maps and Bi-Harmonic Maps on CR-Manifolds and Foliated Riemannian Manifolds. Jour-nal of Applied Mathematics and Physics, 4, 2272-2289. http://dx.doi.org/10.4236/jamp.2016.412219 Received: July 19, 2016 Accepted: December 26, 2016 Published: December 29, 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

Open Access

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The B. Y. Chen’s conjecture: Every biharmonic isometric immersion into the Eucli- dean space n must be harmonic (minimal).

The generalized B. Y. Chen’s conjecture: Every biharmonic isometric immersion of a Riemannian manifold ( ),M g into a Riemannian manifold ( ),N h 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 ( ),M gθ be a complete

strictly pseudoconvex CR manifold, and ( ),N h , a Riemannian manifold of non-positive curvature. Then, every pseudo biharmonic isometric immersion ( ) ( ): , ,M g N hθϕ → 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 ϕ be a pseudo biharmonic map of a strictly pseudoconvex complete CR manifold ( ),M gθ into another Riemannian manifold ( ),N h of non positive curvature.

If ϕ has finite pseudo bienergy ( ),2bE ϕ < ∞ and finite pseudo energy ( )bE ϕ < ∞ , then it is pseudo harmonic, i.e., ( ) 0bτ ϕ = .

Next, let us consider the analogue of harmonic maps and biharmonic maps for folia-tions 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 in-troduced by Konderak and Wolak (cf. [27] [28]).

Among smooth foliated maps ϕ between two Riemannian foliated manifolds, one can define the transversal energy and derive the Euler-Lagrange equation, and trans-versally harmonic map as its critical points which are by definition the transversal ten-sion field vanishes, ( ) 0bτ ϕ = . The transverse bienergy can be also defined as

( ) ( ) 22

12 b gM

E vϕ τ ϕ= ∫ whose Euler-Lagrange equation is that the transversal biten-

sion field ( )2,bτ ϕ vanishes and the transversally biharmonic maps which are, by defi-nition, 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 ( ), ,M g into another Riemannian foliated manifold ( ), ,M g′ ′ ′ of which transversal sectional curvature is non-positive. Then, we will show (cf. [33]) that:

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Theorem 1.2. (cf. Theorem 2.6) Let ( ), ,M g and ( ), ,M g′ ′ ′ be two Riemannian foliated manifolds, and assume that the transversal sectional curvature of ( ), ,M g′ ′ ′ is non-positive. Let ( ) ( ): , , , ,M g M gϕ ′ ′ ′→ be a smooth foliated map which is an isometric immersion of ( ),M g into ( ),M g′ ′ . If ϕ is transversally biharmonic with the finite transversal energy ( ) <E ϕ ∞ and finite transversal bienergy ( )2E ϕ < ∞ , then it is transversally harmonic.

Finally, in Section 5, instead of isometric immersions, we will consider a principal G-bundle ( ) ( ): , ,P g M hπ → , and show a new result whose details will be appeared in [34].

Theorem 1.3. (cf. Theorem 5.1) Let ( ) ( ): , ,P g M hπ → be a principal G-bundle over a Riemannian manifold ( ),M h whose Ricci tensor is negative definite. Then, if

( ) ( ): , ,P g M hπ → 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 ϕ of a Riemannian manifold ( ),M g into another Riemannian manifold ( ),N h , the energy functional ( )E • is defined by

( ) 21: d2 gM

E vϕ ϕ= ∫ (2.1)

whose first variational formula is:

( ) ( )0

d , .d t gM

t

E V vt

ϕ τ ϕ=

= −∫ (2.2)

Here, V is a variational vector field is given by ( ) ( )0

ddx t x

t

V x T Nt ϕϕ

=

= ∈ , ( )x M∈ ,

and the tension field ( )τ ϕ is given by

( ) ( )( )1

: , ,m

i ii

B e eτ ϕ ϕ=

= ∑ (2.3)

( )( ) ( ) ( ) ( ) ( )d, : d d , , ,h gXXB X Y Y Y X Y Mϕϕ ϕ ϕ= ∇ − ∇ ∈X (2.4)

where g∇ and h∇ are Levi-Civita connections of ( ),M g and ( ),N h , respectively. Then, ( ) ( ): , ,M g N hϕ → is harmonic if ( ) 0τ ϕ = .

The second variation formula of the energy functional ( )E • for a harmonic map ( ) ( ): , ,M g N hϕ → is:

( ) ( )2

20

d , ,d t gM

t

E J V V vt

ϕ=

= ∫ (2.5)

where

( ) ( ): ,J V V V= ∆ − (2.6)

( ) ( )( ) ( )1

: , : ,d d ,m

hi i

iV V V R V e eϕ ϕ

∗

=

∆ = ∇ ∇ = ∑ (2.7)

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where 1

mi i

e=

is a locally defined frame field on ( ),M g . The k -energy functional due to J. Eells and L. Lemaire ([1] [2] [3]) is

( ) ( ) ( )21: 1, 2,

2k

k gME d v kϕ δ ϕ= + =∫ (2.8)

which turn out that

( ) ( ) ( ) 221 2

1 1d , .2 2g gM M

E v E vϕ ϕ ϕ τ ϕ= =∫ ∫ (2.9)

Furthermore, the first variation formula for ( )2E ϕ is (cf. [4]):

( ) ( )2 20

d , ,d t gM

t

E V vt

ϕ τ ϕ=

= −∫ (2.10)

( ) ( )( ) ( ) ( )( )2 : .Jτ ϕ τ ϕ τ ϕ τ ϕ= = ∇ − (2.11)

Then, one can define that ( ) ( ): , ,M g N hϕ → is biharmonic (cf. [4]) if ( )2 0τ ϕ = .

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) ( ),M gθ of ( )2 1n + -dimension, and the Webster Riemannian metric gθ given by

( ) ( )( ) ( ) ( ), d , , , 0, , 1g X Y X JY g X T g T Tθ θ θθ= = =

for ( ),X Y H M∈ . Recall the material on the Levi-Civita connection gθ∇ of ( ),M gθ . Due to Lemma 1.3, Page 38 in [35], it holds that,

( ) 2 ,g A T Jθ τ θ θ∇ = ∇ + Ω− ⊗ + ⊗ + (2.12)

where ∇ is the Tanaka-Webster connection, dθΩ = , and ( ) ( ), ,A X Y g X Yθ τ= , ( ),X T T Xτ ∇= , and T∇ is the torsion tensor of ∇. And also, ( )( ) ( ),X Y Y Xτ θ θ τ⊗ = ,

( )( ) ( ) ( ) 1,2

J X Y X JY Y JXθ θ θ= + for all vector fields X, Y on M. Here, J is the

complex structure on ( )H M and is extended as an endomorphism on ( )M by 0JT = .

Then, we have

( ), ,k

gk X k k kX k

X X A X X Tθ∇ = ∇ − (2.13)

0,gT Tθ∇ = (2.14)

where 2

1

nk k

X=

is a locally defined orthonormal frame field on ( )H M with respect to gθ, and T is the characteristic vector field of ( ),M gθ . For (3.6), it follows from that ( ), 0k kX XΩ = , ( )( ), 0k kX Xτ θ⊗ = and ( )( ), 0k kJ X Xθ = since ( ) 0kXθ = .

For (3.7), notice that the Tanaka-Webster connection ∇ satisfies 0TT∇ = , and also 0Tτ = and JT = 0, so that ( ) ( ) ( )( ) ( )( ), 0, , 0, , 0 , 0T T A T T T T J T Tτ θ θΩ = = ⊗ = =

which imply (3.7).

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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 ( ),M gθ be a complete strictly pseudoconvex CR manifold, and assume that ( ),N h is a Riemannian manifold of non-positive curvature.

Then, every pseudo biharmonic isometric immersion ( ) ( ): , ,M g N hθϕ → 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 ( ),M gθ into another Riemannian manifold ( ),N h of non positive curvature.

If φ has finite pseudo bienergy ( ),2bE ϕ < ∞ and finite pseudo energy ( )bE ϕ < ∞ , then it is pseudo harmonic, i.e., ( ) 0bτ ϕ = .

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 mani-folds following [31] [32] [36]. Let ( ), ,M g be an ( )n p q= + -dimensional foliated Riemannian manifold with foliation of codimension q and a bundle-like Rieman-nian 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 Qg the induced Riemannian metric on the normal bundle Q, and Q∇ , the transversal Levi-Civita connection on Q, QR , the transversal curvature tensor, and

QK , the transversal sectional curvature, respectively. Notice that the bundle projection :TM Qπ → is an element of the space ( )1 ,M QΩ of Q-valued 1-forms on M. Then,

one can obtain the Q-valued bilinear form α on M, called the second fundamental form of , defined by

( ) ( )( ) ( ) ( )( ), π π , , .QX XX Y D Y Y X Y Lα = − = ∇ ∈Γ

The trace τ of α , called the tension field of is defined by

( ), 1

, ,p

iji j

i jg X Xτ α

=

= ∑

where 1

pi i

X=

spanns ( )L UΓ on a neighborhood U on M. The Green’s theorem, due to Yorozu and Tanemura [36], of a foliated Riemannian manifold ( ), ,M g says that

( ) ( ) ( )( )div , ,D g Q gM Mv g v Qν τ ν ν= ∈Γ∫ ∫ (2.15)

where ( )div D ν denotes the transversal divergence of ν with respect to Q∇ given by ( ) ( )( ), 1div : ,π

a

q abD Q X ba b g g D Xν ν

== ∑ . Here 1

qa a

X=

spanns ( )L U⊥Γ where

L⊥ is the orthogonal complement bundle of L with a natural identification :Q Lσ≅

⊥→ .

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2.4. The Variational Formulas for Foliations

Let ( ), ,M g , and ( ), ,M g′ ′ ′ be two compact foliated Riemannian manifolds. The transversal energy ( )E ϕ among the totality of smooth foliated maps from ( ), ,M g into ( ), ,M g′ ′ ′ by

( ) 21 .2 T gM

E d vϕ ϕ= ∫ (2.16)

Here, a smooth map ϕ is a foliated map is, by definition, for every leaf L of , there exists a leaf L′ of ′ satisfying ( )L Lϕ ′⊂ . Then, : ;Td d Q Qϕ π ϕ σ′ ′= → can be regarded as a section of 1Q Qϕ∗ − ′⊗ where Q∗ is a subspace of the cotangent bundle T*M. Here, π, π ′ are the projections of TM Q TML→ = and

/TM Q TM L′ ′ ′ ′→ = . 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

tϕ with 0ϕ ϕ= and 0

dd

t

t

Vtϕ

=

= in which V being a section 1Qϕ− ′ ,

( ) ( )0

d , .d t b gM

t

E V vt

ϕ τ ϕ=

= −∫ (2.17)

Here, ( )bτ ϕ is the transversal tension field defined by

( ) ( )( )1

,q

b E T aaa

d Eτ ϕ ϕ=

= ∇∑ (2.18)

where ∇ is the induced connection in 1Q Qϕ∗ − ′⊗ from the Levi-Civita connection of ( ),M g′ ′ , and 1

qa a

E=

is a locally defined orthonormal frame field on Q. Definition 2.2. A smooth foliated map ( ) ( ): , , , ,M g M gϕ ′ ′ ′→ is said to be

transversally harmonic if ( ) 0bτ ϕ ≡ . Then, for a transversally harmonic map ( ) ( ): , , , ,M g M gϕ ′ ′ ′→ , the second

variation formula of the transversal energy ( )E ϕ is given as follows (cf. [?], p. 7) : let

, :s t M Mϕ ′→ ( ),s t− < < be any two parameter smooth foliated variation of ϕ

with ( ) ( )

,

, 0,0

s t

s t

Vsϕ

=

∂=

∂,

( ) ( )

,

, 0,0

s t

s t

Wtϕ

=

∂=

∂ and 0,0ϕ ϕ= ,

( ) ( )( ) ( )

( )

( )

2

,, 0,0

,

Hess , :

, ,

s ts t

b gM

E V W Es t

J V W v

ϕ

ϕ

ϕ=

∂=∂ ∂

= ∫ (2.19)

where ,bJ ϕ is a second order semi-elliptic differential operator acting on the space

( )1Qϕ− ′Γ of sections of 1Qϕ− ′ which is of the form:

( ) ( )

( ) ( )( ) ( )

,

1 1

: trace ,

,a a aEa

Qb Q T T

q qQ

E E E T a T aa a

J V V R V d d

V R V d E d E

ϕ ϕ ϕ

ϕ ϕ

′∗

′∇∇

= =

= ∇ ∇ −

= − ∇ ∇ −∇ −∑ ∑

(2.20)

for ( )1V Qϕ− ′∈Γ . Here, ∇ is the Levi-Civita connection of ( ),M g , and recall also that:

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( )1

: ,a a aa

q

E E EEa

V V∗∇

=

∇ ∇ = − ∇ ∇ −∇∑ (2.21)

( ) ( )( ) ( )1

trace , : , .q

Q QQ T T T a T a

aR V d d R V d E d Eϕ ϕ ϕ ϕ′ ′

=

= ∑ (2.22)

Definition 2.3. The transversal bitension field ( )2,bτ ϕ of a smooth foliated map ϕ is defined by

( ) ( )( )2, ,: .b b bJ ϕτ ϕ τ ϕ= (2.23)

Definition 2.4. The transversal bienergy E2 of a smooth foliated map ϕ is defined by

( ) ( ) 22

1: .2 b gM

E vϕ τ ϕ= ∫ (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 tϕ of ϕ , it holds (cf. [32], p. 13) that

( ) ( )2 2,0

d , .d t b gM

t

E V vt

ϕ τ ϕ=

= −∫ (2.25)

Definition 2.5. A smooth foliated map ( ) ( ): , , , ,M g M gϕ ′ ′ ′→ is said to be transversally biharmonic if ( )2, 0bτ ϕ ≡ .

Then, one can ask the following generalized B.Y. Chen’s conjecture: The generalized Chen’s conjecture: Let ϕ be a transversally biharmonic map from a foliated Riemannian manifold

( ), ,M g into another foliated Riemannian manifold ( ), ,M g′ ′ ′ whose transversal sectional curvature QK ′ is non-positive. Then, ϕ 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 ϕ has both the finite transversal energy and the finite transversal bienergy:

Theorem 2.6. Let ( ) ( ): , , , ,M g M gϕ ′ ′ ′→ a smooth foliated map which is an isometric immersion of ( ),M g into ( ),M g′ ′ . Assume that ( ),M g is complete (possibly non-compact), and the transversal sectional curvature QK ′ of ( ), ,M g′ ′ ′ is non-positive.

If φ is transversally biharmonic having both the finite transversal energy ( )E ϕ < ∞ and the finite transversal bienergy ( )2E ϕ , then it is transversally harmonic.

Remark that in the case that M 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 0x M∈ , let ( ) ( ) 0 :rB x x M r x r= ∈ <

where ( )r x is a distance function on ( ),M gθ , and let us take a cut off function η

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on ( ),M gθ , i.e.,

( ) ( )( ) ( )( )( ) ( )( )

( )

0

2 0

0 1 ,1 ,

0 ,

2 ,

r

r

g

x x Mx x B x

x x B x

x Mr

θ

ηη

η

η

≤ ≤ ∈ = ∈ = ∈/∇ ≤ ∈

(3.1)

where r is the distance function from 0x M∈ , and gθ∇ is the Levi-Civita connection of ( ),M gθ , respectively. Assume that ( ) ( ): , ,M g N hθϕ → is a pseudo biharmonic map, i.e.,

( ) ( )( )

( )( ) ( ) ( )( ) ( ),2

2

1,d d

0.

b b b

nh

b b b j jj

J

R X X

τ ϕ τ ϕ

τ ϕ τ ϕ ϕ ϕ=

=

= ∆ −

=

∑ (3.2)

(The second step) Then, we have

( )( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

2

22

1

, d

,d d , d .

nb b bM

n nhb j j bM

jR X X

τ ϕ η τ ϕ θ θ

η τ ϕ ϕ ϕ τ ϕ θ θ=

∆

=

∫

∑∫

∧

∧ (3.3)

In (3.3), notice that ( ) ( )( ) ( ) ( ),d d ,hb j j bR X Xτ ϕ ϕ ϕ τ ϕ is the sectional curvature

of ( )Nh corresponding to the vectors ( )bτ ϕ and ( )d jXϕ . Since ( ),N h 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

( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

2

2

22

1

, d

, d

, d .

nb b bM

H H nb bM

n nX Xb bj jM

j

τ ϕ η τ ϕ θ θ

τ ϕ η τ ϕ θ θ

τ ϕ η τ ϕ θ θ=

∆ ∧

= ∇ ∇ ∧

= ∇ ∇ ∧

∫

∫

∑∫

(3.4)

Here, let us recall, for ( )1,V W TNϕ−∈Γ , 2

1, , , ,i i

nH H H He e X X

jV W V W V Wα α

α =

∇ ∇ = ∇ ∇ = ∇ ∇∑ ∑

where eα is a locally defined orthonormal frame field of ( ),M gθ and HXW∇

( ) ( )( )1,X M W TNϕ−∈ ∈ΓX is defined by

( ) H H HX Xj j j jj

W X f V f V∇ = + ∇∑

for i jjW f V= ∑ ( )( )jf C M∞∈ and ( )1jV TNϕ−∈Γ . Here, HX is the ( )H M

-component of X corresponding to the decomposition of ( ) ( )x x xT M H M T= ⊕ ( )x M∈ , and ∇ is the induced connection of 1TNϕ− from the Levi-Civita con- nection h∇ of ( ),N h .

Since

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( )( ) ( ) ( )2 22 ,j jX Xb j b bXη τ ϕ η ητ ϕ η τ ϕ∇ = + ∇ (3.5)

the right hand side of (3.4) is equal to

( ) ( ) ( ) ( ) ( ) ( )2 222

1 1d 2 , d .j j

n nn nX Xb b j bM M

j jXη τ ϕ θ θ η τ ϕ η τ ϕ θ θ

= =

∇ + ∇∑ ∑∫ ∫∧ ∧ (3.6)

Therefore, together with (3.3), we have

( ) ( )

( ) ( ) ( ) ( )

( )

2 22

1

2

1

2

1

d

2 , d

: 2 , d ,

j

n nX bM

j

n nX b j bjM

j

n nj jM

j

X

V W

η τ ϕ θ θ

η τ ϕ η τ ϕ θ θ

θ θ

=

=

=

∇

≤ − ∇

= −

∑∫

∑∫

∑∫

∧

∧

∧

(3.7)

where we define ( ) ( )1, 1, , 2j jV W TN j nϕ−∈Γ = by

( ) ( ) ( ): , : .jXj b j j bV W Xη τ ϕ η τ ϕ= ∇ =

Then, it holds that 210 i iV W≤ ±

for every 0> which implies that

2 212 , .i i i iV W V W≤ +

Therefore, we have that The right hand side of (3.7)

( ) ( )2 22 2

1 1

1d dn nn n

j jM Mj j

V Wθ θ θ θ= =

≤ +∑ ∑∫ ∫

∧ ∧ (3.8)

foe every 0> . By taking 12

= , we obtain

( ) ( )

( ) ( ) ( ) ( )

2 22

1

2 22 2 22

1 1

d

1 d 2 d .2

j

j

n nX bM

j

n nn nX b j bM M

j jX

η τ ϕ θ θ

η τ ϕ θ θ η τ ϕ θ θ

=

= =

∇

≤ ∇ +

∑∫

∑ ∑∫ ∫

∧

∧ ∧

(3.9)

Therefore, we obtain, due to the properties that 1η = on ( )0rB x , and 2

2221

2n gjj X

rθη η

=

≤ ∇ ≤

∑ ,

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

0

2 22 22

1 1

2 2 2

1

2

2

d d

4 d

16 d .

jr

n nn nXb bB x M

j j

n nj bM

j

nbM

X

r

τ ϕ θ θ η τ ϕ θ θ

η τ ϕ θ θ

τ ϕ θ θ

= =

=

∇ ≤ ∇

≤

≤

∑ ∑∫ ∫

∑∫

∫

∧ ∧

∧

∧

(3.10)

(The third step) By our assumption that ( ) ( ) ( )2,2

12

nb bM

E dϕ τ ϕ θ θ= < ∞∫ ∧ and

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( ),M gθ is complete, if we let r →∞ , then ( )0rB x goes to M, and the right hand side of (3.10) goes to zero. We have

( ) ( )2 2

1d 0.j

n nX bM

jτ ϕ θ θ

=

∇ =∑∫ ∧ (3.11)

This implies that

( ) ( )( )0 for all .X b X H Mτ ϕ∇ = ∈ (3.12)

(The fourth step) Let us take a 1 form α on M defined by

( )( ) ( ) ( )( )

( )d , , ,

0, .

bX X H MX

X T

ϕ τ ϕα

∈= =

Then, we have

( ) ( ) ( )

( )( ) ( ) ( )( )( ) ( )

12 22

1

1122 22

,2

d d

d d

2 < ,

nn njM M

j

n nb bM

b b

X

d

E E

α θ θ α θ θ

ϕ θ θ τ ϕ θ θ

ϕ ϕ

=

=

≤

= ∞

∑∫ ∫

∫

∧ ∧

∧ ∧ (3.13)

where we put ( )21: dn

b i iid X Xϕ ϕ=

= ⊗∑ ,

( ) ( ) ( )( ) ( ) ( )( )2 22

, 1 1, d ,d d ,d ,

n n

b i j i j i ii j i

d g X X h X X h X Xθϕ ϕ ϕ ϕ ϕ= =

= =∑ ∑

and

( ) ( )21 d .2

nb bM

E dϕ ϕ θ θ= ∫ ∧ (3.14)

Furthermore, let us define a C∞ function bδ α on M by

( )( ) ( )( ) ( ) 2 2

1 1,

j

n n

b X j j j X jjj j

X X X Xδ α α α α= =

= − ∇ = − − ∇∑ ∑ (3.15)

where ∇ is the Tanaka-Webster connection. Notice that

( ) ( )( ) ( )( )

( )( ) ( ) ( )( ) ( )

( )( ) ( )( ) ( )( ) ( )

2

1

2

1

2

1

2

1

div

(

,

j

j

j

j

ng g

j TXj

ng g

j H j H j H H TXj

ng

j j H jXj

n

j j X jj

b

X T

X X X T T T

X X X

X X X

θ θ

θ θ

θ

α α α

α π α π α π α π

α α π

α α

δ α

=

=

=

=

= ∇ + ∇

= − ∇ + − ∇

= − ∇

= − ∇

= −

∑

∑

∑

∑

(3.16)

where ( ) ( ):H x xT M H Mπ → is the natural projection. We used the facts that 0g

T Tθ∇ = , and ( )gH X XY Yθπ ∇ = ∇ ( )( ),X Y H M∈ ([35], p.37). Here, recall again

gθ∇ is the Levi-Civita connection of gθ , and ∇ is the Tanaka-Webster connection.

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Then, we have, for (3.16),

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( )

2

1

2

1

2

1

2

d , d ,

d , d , d ,

d d ,

.

j

j j j

j j

n

b j j b X j bj

n

X Xj b j b X j bj

n

X j X j bj

b

X X X

X X X

X X

δ α ϕ τ ϕ ϕ τ ϕ

ϕ τ ϕ ϕ τ ϕ ϕ τ ϕ

ϕ ϕ τ ϕ

τ ϕ

=

=

=

= − − ∇

= − ∇ + ∇ − ∇

= − ∇ − ∇

= −

∑

∑

∑

(3.17)

We used (3.12) ( ) 0jX bτ ϕ∇ = to derive the last second equality of (3.17). Then, due to (3.17), we have for ( ),2bE ϕ ,

( ) ( ) ( )

( )

( ) ( )

2,2

1 d2

1 d2

1 div d20.

nb bM

nbM

n

M

E ϕ τ ϕ θ θ

δ αθ θ

α θ θ

=

= −

=

=

∫

∫

∫

∧

∧

∧

(3.18)

In the last equality, we used Gaffney’s theorem ([16], p. 271, [?]). Therefore, we obtain ( ) 0bτ ϕ ≡ , i.e., ϕ 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 η from a fixed point 0x M∈ on ( ),M g , i.e.,

( ) ( )( ) ( )( )( ) ( )( )

( )

0

2 0

0 1 ,1 ,

0 ,

2 ,

r

r

g

x x Mx x B x

x x B x

x Mr

ηη

η

η

≤ ≤ ∈ = ∈ = ∉∇ ≤ ∈

where ( ) ( ) 0 :rB x x M r x r= ∈ < , ( )r x is a distance function from 0x on ( ),M g , g∇ is the Levi-Civita connection of ( ),M g , respectively. Assume that ϕ is a transversally biharmonic map of ( ), ,M g F into ( ), ,M g F′ ′ ′ ,

i.e.,

( ) ( )( )( ) ( )( )

2, ,

trace ,

0,

b b b

Qb Q b T T

J J

R d dϕϕ τ ϕ

τ ϕ τ ϕ ϕ ϕ′∗

=

= ∇ ∇ −

=

(4.1)

where recall ∇ is the induced connection on 1Q T Mϕ− ∗′⊗ . (The second step) Then, by (4.1), we obtain that

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( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( )

2 2

2

1

2,

1

, trace , ,

, ,

0,

Qb b g Q b T T b gM M

qQ

b T a T a b gMaq

Qa gM

a

v R d d v

R d E d E v

K vϕ

τ ϕ η τ ϕ η τ ϕ ϕ ϕ τ ϕ

η τ ϕ ϕ ϕ τ ϕ

η

′∗

′

=

′

=

∇ ∇ =

=

= Π

≤

∫ ∫

∑∫

∑∫

(4.2)

where the sectional curvature ( ),Q

aK ϕ′ Π of ( ), ,M g′ ′ ′ corresponding to the plane

spanned by ( )bτ ϕ and ( )T ad Eϕ is non-positive. (The third step) On the other hand, the left hand side of (4.2) is equal to

( ) ( )( )

( ) ( )( )

( ) ( ) ( ) ( )

2

2

1

22

1 1

,

,

2 ,

a a

a a

b b gM

q

E b E b gMa

q q

E b g E b a b gM Ma a

v

v

v E v

τ ϕ η τ ϕ

τ ϕ η τ ϕ

η τ ϕ η τ ϕ η τ ϕ

=

= =

∇ ∇

= ∇ ∇

= ∇ + ∇

∫

∑∫

∑ ∑∫ ∫

(4.3)

since

( )( ) ( ) ( ) ( )2 2 2 .a aE b E b a bEη τ ϕ η τ ϕ η η τ ϕ∇ = ∇ +

Together (4.2) and (4.3), we obtain

( ) ( ) ( ) ( )

( ) ( )

22

1 1

2 222

1 1

2 ,

1 2 .2

b a

a

q q

g E b a b gM Ma a

q q

E b g a b gM Ma a

v E v

v E v

η τ ϕ η τ ϕ η τ ϕ

η τ ϕ η τ ϕ

= =

= =

∇ ≤ − ∇

≤ ∇ +

∑ ∑∫ ∫

∑ ∑∫ ∫

(4.4)

Because, putting ( ):aa E bV η τ ϕ= ∇ , ( ) ( ):a a bW E η τ η= ( )1, ,a q= , we have

22 21 10 2 ,a a a a a aV W V V W W≤ ± = ± +

which is

2 212 , .a a a aV W V W≤ +

(4.5)

If we put 12

= in (4.5), then we obtain

( )2 212 , 2 1, , .2a a a aV W V W a q≤ + = (4.6)

By (4.6), we have the second inequality of (4.4).

(The fourth step) Noticing that η = 1 on ( )0rB x and 2 2aE

rη ≤ in the inequality

(3.4), we obtain

( ) ( ) ( ) ( ) ( )

( ) ( )

2 2 22 2

0 01 1 1

2 222

1

164 .

a a a

q q q

E b g E b g E b gB x B x Mr ra a aq

a b g b gM Ma

v v v

E v vr

τ ϕ η τ ϕ η τ ϕ

η τ ϕ τ ϕ

= = =

=

∇ = ∇ ≤ ∇

≤ ≤

∑ ∑ ∑∫ ∫ ∫

∑∫ ∫

(4.7)

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2284

Letting r →∞ , the right hand side of (4.7) converges to zero since

( ) ( ) 22

12 b gM

E vϕ τ ϕ= < ∞∫ . But due to (4.7), the left hand side of (4.7) must converge

to ( )2

1 a

qE b gaM

vτ ϕ=∇∑∫ since ( )0rB X tends to M because ( ),M g is complete.

Therefore, we obtain that

( )2

10 0,

a

q

E b gMa

vτ ϕ=

≤ ∇ ≤∑∫

which implies that

( ) ( ) ( ) ( )( )0 1, , , . ., 0 .aE b X ba q i e X Qτ ϕ τ ϕ∇ = = ∇ = ∀ ∈Γ

(4.8)

(The fifth step) Let us define a 1-form α on M by

( ) ( )( ) ( ) ( )( ): d , , ,bX X X Mα ϕ π τ ϕ= ∈X (4.9)

and a canonical dual vector field ( )# Mα ∈X on M by ( ) ( )( )# , : , Y Y Y Mα α= ∈X . Then, its divergence ( )#div α written as

( ) ( ) ( )# # #1 1div , ,

i a

p qg gE i E ai ag E g Eα α α

= == ∇ + ∇∑ ∑ ,

can be given as follows. Here, 1

pi i

E=

and 1

qa a

E=

are locally defined orthonormal frame fields on leaves L of and Q, respectively, ( dim xL p= , dim xQ q= , x M∈ ).

Then, we can calculate ( )#div α as follows:

( ) ( )( ) ( ) ( )( ) ( )

( )

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )( ) ( )

#

1 1

1

1

1

1

1

div

d ,

d , d ,

d ,

d , d , d ,

d

i a

i

a

i

a

i

p qg g

i i E i a a E ai a

pgE i b

i

qg

a a b E a ba

pgE i b

i

qg

E a b a E b E a ba aa

pgE i

i

E E E E E E

E

E E E

E

E E E

E

α α α α α

ϕ π τ ϕ

ϕ τ ϕ ϕ π τ ϕ

ϕ π τ ϕ

ϕ τ ϕ ϕ τ ϕ ϕ π τ ϕ

ϕ π

= =

=

=

=

=

=

= − ∇ + − ∇

= − ∇

+ − ∇

= − ∇

+ ∇ + ∇ − ∇

= − ∇

∑ ∑

∑

∑

∑

∑

∑

( )( ) ( )( ) ( )1

d d , .a a

qg

E a E a ba

E Eϕ ϕ π τ ϕ=

+ ∇ − ∇

∑

(4.10)

since ( ) 0E baτ ϕ∇ = in the last equality of (4.10). Integrating the both hands of (4.10)

over M, we have

( )

( )( ) ( )( ) ( )

1

1

d ,

d d , .

i

a

pgE i b gM

i

qg

E a E a b gaMa

E v

E E v

ϕ π τ ϕ

ϕ ϕ π τ ϕ

=

=

∇

= ∇ − ∇

∑∫

∑∫

(4.11)

because of ( )#div 0gMvα =∫ . Notice that both hands in (4.11) are well defined because

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2285

of ( )E ϕ < ∞ and ( )2E ϕ < ∞ .

Since ( )#1:

i

p gE ii Eκ π

== ∇∑ is the second fundamental form of each leaf L in ( ),M g

and

( ) ( )( ) ( )

( )( ) ( )( ) 1

1 1

d d

d d d ,

a a

a a a

qg

b E a E aa

q qg g

E a E a E aa a

E E

E E E

τ ϕ ϕ ϕ

ϕ ϕ π ϕ

=

⊥

= =

= ∇ − ∇

= ∇ − ∇ − ∇

∑

∑ ∑

(4.12)

the right hand side of (4.11) coincides with

( ) ( )1

d , ,a

qg

b E a b gMa

E vτ ϕ ϕ τ ϕ⊥

=

+ ∇ ∑∫ (4.13)

(4.11) is equivalent to that

( ) ( ) ( ) ( ) ( )#

1d , , d , .

a

qg

b g b b g E a b gM M Ma

v v E vϕ κ τ ϕ τ ϕ τ ϕ ϕ τ ϕ⊥

=

= + ∇ ∑∫ ∫ ∫ (4.14)

If ( ) ( ): , ,M g M gϕ ′ ′→ is an isometric immersion, then it holds that ( ) ( )d , 0x x bT Mϕ τ ϕ = , which implies that both the left hand side and the second term

of the right hand side of (4.14) vanish, that is, ( ) ( ), 0b b gMvτ ϕ τ ϕ =∫ . Therefore

( ) 0bτ ϕ ≡ . 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 ( ) ( ): , ,P g M hπ → be a principal G-bundle over a Riemannian manifold ( ),M h whose Ricci tensor is negative definite. Then, if ( ) ( ): , ,P g M hπ → is biharmonic, then it is harmonic.

Let us consider a principal G-bundle ( ) ( ): , ,P g M hπ → whose the total space P is compact. Assume that the projection ( ) ( ): , ,P g M hπ → is biharmonic, which is by definition, ( )( ) 0J τ π ≡ , where ( )τ π is the tension field of π which is defined by

( ) ( ) 1

: ,ei i

ph ge i i

ie eτ π π π∗ ∗

=

= ∇ − ∇∑ (5.1)

the Jacobi operator J is defined by

( ) ( )( )1: π ,JV V V V TN−= ∆ − ∈Γ (5.2)

∆ is the rough Laplacian defined by

( ) 1

: ,gi i ei

p

e e eii

V V V∇=

∆ = − ∇ ∇ −∇∑ (5.3)

and

( ) ( ): , ,hi iV R V e eπ π∗ ∗= (5.4)

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2286

where 1

pi i

e=

is a locally defined orthonormal frame field on ( ),P g . The tangent space uP ( )u P∈ is canonically decomposed into the orthogonal di-

rect sum of the vertical subspace u uG A A∗= ∈g and the horizontal subspace uH :

u u uP G H= ⊕ . Then, we have

( ) ( ) ( )( )

( ) ( )( ) ( )( )

( ) ( )( )

21

1 1

1

,

, ,

, ,

ph

i iim k

h hi i m i m i

i im

hi i

i

R e e

R e e R A A

R e e

τ π τ π τ π π π

τ π τ π π π τ π π π

τ π τ π π π

∗ ∗=

∗ ∗∗ ∗ ∗ + ∗ +

= =

∗ ∗=

= ∆ −

= ∆ − −

= ∆ −

∑

∑ ∑

∑

where dim , dim , dimp P m M k G= = = , respectively. Then, we obtain

( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

1

1

0 ,

, , ,

, , ,

gMm

hg i i gM M

im

hg i i gM M

i

J v

v R e e v

v R e e v

τ π τ π

τ π τ π τ π π π τ π

τ π τ π τ π π π τ π

∗

∗ ∗=

∗ ∗=

=

= ∇ ∇ −

= ∇ ∇ −

∫

∑∫ ∫

∑∫ ∫

Therefore, we obtain

( ) ( ) ( )( ) ( )

( )( )

( )( )

1

1

, , ,

,

Ric ,

mh

g i i gM Mim

hiM

i

hgM

v R e e v

K e

v

τ π τ π τ π π π τ π

τ π π

τ π

∗ ∗=

∗=

∇ ∇ =

=

=

∑∫ ∫

∑∫

∫

(5.5)

where we denote by ( ),hK u v ( ), xu v T M∈ , the sectional curvature of ( ),M h through two plane of xT M given by , xu v T M∈ , and ( )Ric X is the Ricci curvature of ( ),M h along xX T M∈ . 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 ( ),M h is non-positive. Therefore, we obtain

( ) ( )( ) ( )( )( )

0 , . ., is parallel, and

Ric 0.X

h

X P i eτ π τ π

τ π

∇ = ∀ ∈

=

X (5.6)

Let us consider a 1-form α on M defined by

( ) ( )( ) ( )( ) ( ) ( )( ): , , , .xY x x Y Y M x Pπα π τ π= ∈ ∈X (5.7)

Then, for every ( ),Y Z M∈X , we have

( )( ) ( )( ) ( )( ) ( )

( ) ( ) ( )

, ,

, , ,

0,

h hZ Z

hZ

h hZ Z Z

Y Z Y Y

Z Y Y

Y Y Y

α α α

τ π τ π

τ π τ π τ π

∇ = − ∇

= − ∇

= ∇ + ∇ − ∇

=

(5.8)

which implies that α is parallel 1-form on ( ),M h . Since we assume that the Ricci

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2287

tensor of ( ),M h is negative definite, α must vanish (so called Bochner’s theorem, cf. [40], p. 55). Therefore, ( ) 0τ π ≡ , i.e., the projection of the principal G-bundle

( ) ( ): , ,P g N hπ → is harmonic. We obtain Theorem 5.1.

Acknowledgements

None.

Funding

Supported by the Grant-in-Aid for the Scientific Research, (C) No. 25400154, Japan So-ciety for the Promotion of Science.

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