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Laplacians on the tangent bundle of Finsler manifold
Chunping Zhong

Abstract. Let M be a smooth manifold with a Finsler metric F and
g̃ be the naturally induced Riemann metric on the slit tangent bundle
M̃ . The Weitzenböck formulas of the horizontal Laplacian △h and the
vertical Laplacian △v are obtained in terms of the Cartan connection of
(M, F ). The relationship between the Hodge-Laplace operator △ of g̃
and the operators △h , △v , △mix are investigated. As application, the
relationship between the eigenvalues of △ and △h , △v are established,
and a Bochner-type vanishing theorem of horizontal differential form on
M̃ is obtained.
M.S.C. 2010: 53C40, 53C60.
Key words: Finsler manifold; horizontal Laplacian; vertical Laplacian.

1

Introduction

Let M be a real manifold of dimension m, we denote by T M its tangent bundle, and
π : T M → M the canonical projection, the cotangent bundle of M is denoted by

T ∗ M . We assume that M is endowed with a Finsler metric F in the sense of [1], see
also [3] and [7]. It is known that there is not a canonical way to define Laplacian on
(M, F ), we refer to [2] for more details. Usually Laplacians on Finsler manifolds are
constructed either on the base manifold M or the slit tangent bundle M̃ = T M − {o},
where o denotes the zero section of T M . In [8], Munteanu obtained the Weitzenböck
formulas of horizontal and vertical Laplacians for 1-forms on a Riemann vector bundle
p : E → M with compact fiber spaces F = p−1 (x) for x ∈ M . Very recently, this
idea was developed in [10] to investigate vanishing theorem of holomorphic forms
on Kähler Finsler manifolds. Note that since most curvature components of Finsler
metrics depend not only on base point x ∈ M but also on direction y ∈ Tx M [9], and
a Finsler metric F on M naturally induces a Riemann metric g̃ on M̃ , it is natural
to define a Laplacian on the tangent bundle T M of a Finsler manifold (M, F ). The
purpose of this paper is to introduce the horizontal and vertical Laplacians of a Finsler
metric on the slit tangent bundle M̃ and relate it to the Hodge-Laplace operator △
of the Riemann metric g̃ on M̃ . It is known that △ is not a type preserving operator
when acting on differential forms of type (p, q) on M̃ , i.e., those forms which are
∗
Balkan
Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 170-181.
c Balkan Society of Geometers, Geometry Balkan Press 2011.

°

Laplacians on the tangent bundle of Finsler manifold

171

horizontal type of p and vertical type of q on M̃ . Nevertheless, there are three
components of △, i.e., the horizonal Laplacian △h , the vertical Laplacian △v and
the mixed part △mix , which are type preserving [8]. Our results shows that both
the local expressions of △h and △v depend on the choice of the Finsler connection
associated to (M, F ). Moreover, the horizontal Laplacian △h depends only on the
horizontal curvature components of F , the vertical Laplacian △v depends only on
the vertical curvature components of F , the mixed Laplacian △mix depends only on
the Riemann curvature of F and is independent of the choice of Finsler connections
associated to (M, F ). We obtain the Weitzenböck formula of △h and △v in terms
of the Cartan connection of (M, F ), and obtain a Bochner-type vanishing theorem of
horizontal differential forms which are compactly supported in M̃ .

2

Preliminaries

In this section, we shall recall some basic facts of Finsler manifold. We refer to [1] for
more details. Let F be a Finsler metric on a real manifold M of dimension m. Denote
by V the vertical vector bundle of M , then there is a Riemann metric g on V which
is induced by F , and a unique good vertical connection ∇ : X (V) → X (T ∗ M̃ ⊗ V)
called the Cartan connection associated to (M, F ). Note that associated to ∇, there
are horizontal bundle H and the horizontal map Θ : V → H. Using Θ one can
transfers the natural Riemann metric g on V to H, and get a Riemann metric g̃
on the whole bundle T M̃ , and consequently a linear connection, still denoted by ∇,
on X (T M̃ ) and X (T ∗ M̃ ). Note that one can also consider the Berwald connection
associated to (M, F ), and it was shown in [5] that the Berwald connection and the
Cartan connection play a different role when one considers the properties of adapted
coordinates system related to them.
Locally let (xi ) = (x1 , · · · , xm ) be the local coordinates on M and (xi , y i ) =
1
(x , · · · , xm , y 1 , · · · , y m ) be the naturally induced coordinates on T M . Denote by
{δi , ∂˙i } the local frame for T M̃ and {dxi , δy i } the dual frame for T ∗ M̃ , here
δi =

∂
∂
− Γ;ij j ,
∂xi
∂y

∂
∂˙i =
,
∂y i

δy i = dy i + Γ;ji dxj ,

and Γ;ij is the nonlinear connection coefficients associated to the Cartan connection
∇. It is known that the Cartan connection ∇ satisfy
∇δk ∂˙j

=

i ˙

i ˙
i
i
Γj;k
∂i , ∇∂˙k ∂˙j = Cjk
∂i , ∇δk δj = Γj;k
δi , ∇∂˙k δj = Cjk
δi ,

(2.2) ∇δk dxi

=

i
−Γj;k
dxj ,

i
∇δk δy i = −Γj;k
δy j ,

(2.3) ∇∂˙k dxi

=

i
−Cjk
dxj ,

i
∇∂˙k δy i = −Cjk
δy j ,

(2.1)

i
i
where in the above equations, Γj;k
and Cjk
are the horizontal and vertical connection

coefficients of the Cartan connection. That is,

(2.4)

i
Γj;k
=

1 hi
1
i
g [δj (ghk ) + δk (gjh ) − δh (gjk )], Cjk
= g hi ∂˙j (ghk ),
2
2

with ghk = 12 ∂˙h ∂˙k (F 2 ). It is clear that
(2.5)

i

i
Γj;k
= Γk;j
,

i
i
Cjk
= Ckj
.

172

Chunping Zhong

Note that {dxi , δy i } is a local frame for T ∗ M̃ , it follows that the differential forms
on M̃ can be represented in terms of {dxi , δy i }. In [8], a differential form ϕ of type
(p, q) on M̃ is expressed in terms of {dxi , δy i } as follows:
(2.6)

ϕ=

1
ϕI J dxIp ∧ δy Jq
p!q! p q

where Ip = i1 · · · ip , Jq = j1 · · · jq , dxIp = dxi1 ∧ · · · ∧ dxip , δy Jq = δy j1 ∧ · · · ∧ δy jq , and
ϕIp Jq are anti-symmetric in their indexes in Ip and Jq , respectively. In the following,
we denote by ∧p,q
c (M̃ ) the space of differential forms on M̃ which are of type (p, q) and
the coefficients ϕIp Jq are compactly supported in M̃ . Note that there is a naturally
point-wise inner product h·, ·i defined in ∧p,q
c (M̃ ) for every (x, y) ∈ M̃ . More precisely,
for ϕ, ψ ∈ ∧p,q
(
M̃
),
c
hϕ, ψi =

1
ϕi ···i j ···j ψr ···r s ···s g i1 r1 · · · g ip rp g j1 s1 · · · g jq sq ,
p!q! 1 p 1 q 1 p 1 q

where the above equation is evaluated at (x, y) ∈ M̃ . The point-wise inner product
h·, ·i gives rise to a global inner product (·, ·) defined in ∧p,q
c (M̃ ). That is,
Z
hϕ, ψidV
(2.7)
(ϕ, ψ) =
M̃

whenever the integral of (2.7) exists. Here dV = det(gij )dx1 ∧· · ·∧dxn ∧dy 1 ∧· · ·∧dy n
is the natural volume form of the Riemann metric g̃ on M̃ . Note that
df = δi (f )dxi + ∂˙i (f )δy i

(2.8)
for f ∈ C ∞ (M̃ ) and

1 i
d(δy i ) = − Rjk
dxj ∧ dxk − Gijk dxj ∧ δy k ,
2
i
i
i
where Rjk
= δk (Γ;ji ) − δj (Γ;k
) and Gijk = ∂˙j (Γ;k
). Thus for ϕ ∈ ∧p,q
c (M̃ ) we
p+1,q
p,q+1
have dϕ = dh ϕ + dv ϕ + dmix ϕ, where dh ϕ ∈ ∧c
(M̃ ), dv ϕ ∈ ∧c
(M̃ ) and
dmix ϕ ∈ ∧cp+2,q−1 (M̃ ). Let d∗h , d∗v and d∗mix be the formal adjoints of dh , dv and
dmix , respectively with respect to the global inner product (2.7), i.e.,

(2.9)

(2.10) (dh ϕ, ψ) = (ϕ, d∗h ψ), (dv ϕ, ψ) = (ϕ, d∗v ψ), (dmix ϕ, ψ) = (ϕ, d∗mix ψ).
Let i be the substitution operator and e be the wedge product operator. The
operators i and e satisfy
(2.11)

i(δj )dxk = δjk , i(δj )δy k = 0, i(∂˙j )dxk = 0, i(∂˙j )δy k = δjk ,

(2.12)

e(dxi )ϕ = dxi ∧ ϕ, e(δy i )ϕ = δy i ∧ ϕ

for ϕ ∈ ∧p,q
c (M̃ ), and are anti-commutative, i.e.,
(2.13)

i(X)i(Y )ϕ = −i(Y )i(X)ϕ,

e(φ)e(ω)ϕ = −e(ω)e(φ)ϕ

for X, Y ∈ X (T M̃ ) and φ, ω ∈ X (T ∗ M̃ ). In the calculation of dh , dv , dmix and their
formal adjoint d∗h , d∗v , d∗mix in terms of the Cartan connection ∇, equalities (2.2)-(2.3)
and (2.11)-(2.13) will be used repeatedly without explicit statement.

Laplacians on the tangent bundle of Finsler manifold

3

173

Levi-Civita connection of g̃

Lemma 3.1. Let D be the Levi-Civita connection of the Riemann metric g̃ on M̃ .
Then in terms of the local frame {δi , ∂˙i } and its dual frame {dxi , δy i }, we have
³
1 i ´˙
i
i
(3.1)
∂i ,
+ Rjk
Dδk δj = Γj;k
δi − Cjk
2
³
´
1
i
s
i ˙
Dδk ∂˙j =
gsj δi + Γj;k
Cjk
+ g li Rlk
(3.2)
∂i ,
2
³
´
1
i
s
Cjk
+ g li Rlk
(3.3)
gsj δi − Lijk ∂˙i ,
D∂˙j δk =
2
i ˙
D∂˙k ∂˙j = Lijk δi + Cjk
(3.4)
∂i ,
´
³
1
s
i
i
(3.5)
gsj δy j ,
Dδk dxi = −Γj;k
dxj − Cjk
+ g li Rlk
2
³
1 i ´ j
i
i
i
Dδk δy =
Cjk + Rjk dx − Γj;k
(3.6)
δy j ,
2
³
´
1
i
s
D∂˙k dxi = − Cjk
+ g li Rlj
gsk dxj − Lijk δy j ,
(3.7)
2
i
D∂˙k δy i = Lijk dxj − Cjk
(3.8)
δy j .
Proof. We only need to prove (3.1)-(3.4) since one can obtain (3.5)-(3.8) by the following formula
DX ω(Y ) = X(ω(Y )) − ω(DX Y ),

∀X, Y ∈ X (T M̃ ), ω ∈ X (T ∗ M̃ ).

As well known, the Levi-Civita connection D of g̃ is characterized by the following
Koszul formula
2g̃(DX Y, Z) = X g̃(Y, Z)+Y g̃(Z, X)−Z g̃(X, Y )+g̃([X, Y ], Z)−g̃([Y, Z], X)+g̃([Z, X], Y ).
Using the facts that
g̃(δi , δj ) = gij ,

g̃(∂˙i , ∂˙j ) = gij ,

g̃(δi , ∂˙j ) = 0,

we have
(3.9)

Dδk δj

(3.10) Dδk ∂˙j
(3.11) D∂˙k ∂˙j

³
1 i ´˙
i
i
= Γj;k
∂i ,
+ Rjk
δi − Cjk
2
i
´
³
1 h
1
s
i
gsj δi + g li δk (gjl ) + gtl Gtkj − gtj Gtkl ∂˙i ,
=
Cjk
+ g li Rlk
2
2
i
1 li h
s
i ˙
∂i .
= − g δl (gkj ) − gsk Gjl − gsj Gskl δi + Cjk
2

Since gtl Gtkj − gtj Gtkl = δj (glk ) − δl (gjk ), it follows that
(3.12)

i 1 h
i
1 li h
i
.
g δk (gjl ) + gtl Gtkj − gtj Gtkl = g li δk (gjl ) + δj (glk ) − δl (gjk ) = Γj;k
2
2

i
Next, if we denote by gjl;k := δk (gjl ) − gtl Gtjk − gjt Gtlk and Lijk := Gijk − Γj;k
. Then
it is easy to check that

(3.13)

gjl;k = gjk;l ,

Lijk = Likj ,

174

Chunping Zhong

consequently we have
(3.14)

i
1 h
1
Lijk = − g li gjl;k = − g li δl (gkj ) − gsk Gsjl − gsj Gskl .
2
2

This completes (3.2) and (3.4). By the torsion-freeness of D, we get (3.3).

4

¤

Laplacians in terms of Cartan connection

Let d be the exterior differential operator on M̃ and d∗ be the formal adjoint of d
with respect to the global inner product (·, ·) defined by (2.7). In this section we
shall first give a representation of d and d∗ in terms of the horizontal and vertical
covariant derivatives of the Cartan connection ∇. Then we shall derive the horizontal
Laplacian △h and the vertical Laplacian △v in terms of ∇, respectively.
Lemma 4.1. Let (M, F ) be a real Finsler manifold with the Cartan connection ∇.
Then
d =e(dxs )∇δs − Ltrs e(dxr )e(δy s )i(∂˙t )
(4.1)

1 t
t
+ e(δy s )∇∂˙s − Crs
e(dxr )e(δy s )i(δt ) − Rrs
e(dxr )e(dxs )i(∂˙t ).
2

Proof. It is easy to check that (4.1) holds for every ϕ ∈ ∧p,q
c (M̃ ).

¤

k
Lemma 4.2. Let Lkji = Gkji − Γj;i
. Then g ki Lhji = g hi Lkji .

Proof. In deed, it follows from (3.13) and (3.14) that
1
1
1
g hi Lkji = − g hi g lk gjl;i = − g hl g ik gji;l = − g ki g lh gjl;i = g ki Lhji .
2
2
2
¤
Lemma 4.3. Let (M, F ) be a real Finsler manifold with the Cartan connection ∇.
Let ∇ also denote the induced linear connection on T M̃ and T ∗ M̃ , respectively. Then
for every ϕ ∈ ∧p,q
c (M̃ ),
(4.2)
(4.3)
(4.4)
(4.5)

∇δs i(δk )ϕ =
∇δs i(∂˙h )ϕ =

l
i(δl )ϕ,
i(δk )∇δs ϕ + Γk;s
l
˙
i(∂h )∇δ ϕ + Γ i(∂˙l )ϕ,

∇∂˙s i(δk )ϕ =
∇∂˙s i(∂˙h )ϕ =

l
i(δk )∇∂˙s ϕ + Cks
i(δl )ϕ,
l
i(∂˙h )∇ ˙ ϕ + C i(∂˙l )ϕ,

s

∂s

h;s

hs

(4.6)

∇δs e(dx )ϕ =

k
e(dxl )ϕ,
e(dx )∇δs ϕ − Γl;s

(4.7)

∇δs e(δy k )ϕ =

k
e(δy l )ϕ,
e(δy k )∇δs ϕ − Γl;s

(4.8)

∇∂˙s e(dxk )ϕ =

k
e(dxk )∇∂˙s ϕ − Cls
e(dxl )ϕ,

(4.9)

∇∂˙s e(δy k )ϕ =

k
e(δy l )ϕ.
e(δy k )∇∂˙s ϕ − Cls

k

k

Proof. This is a direct calculation by using (2.2)-(2.3) and (2.11)-(2.13).

¤

Laplacians on the tangent bundle of Finsler manifold

175

Lemma 4.4. Let (M, F ) be a real Finsler manifold with the Cartan connection ∇,
and d∗ be the formal adjoint of the exterior differential operator d on M̃ with respect
to the inner product (·, ·) defined by (2.7). Then

(4.10)

d∗ = − g ki i(δk )∇δi + g ki Lhji e(δy j )i(δk )i(∂˙h ) + g ki Lhik i(δh )
³
1 h´ ˙
h
h
+ Rki
− g ki i(∂˙k )∇∂˙i − g ki Cki
i(∂h ) + g ki Cji
e(dxj )i(δk )i(∂˙h )
2
1
s
− g ki g lh Rli
gsj e(δy j )i(δk )i(δh ).
2

Proof. In terms of the local frame {dxi , δy i } for T ∗ M̃ , the dual operator d∗ of d can
be expressed in terms of the Levi-Civita connection D of g̃ in an invariant form:
d∗ ϕ = −g ki i(δk )Dδi ϕ − g ki i(∂˙k )D∂˙i ϕ, ϕ ∈ ∧p,q
c (M̃ ).

(4.11)

Using this and (3.5)-(3.8) and Lemma 4.2, one gets (4.10).

¤

Theorem 4.5. Let (M, F ) be a real Finsler manifold with the Cartan connection ∇.
Let ϕ be a differential form of type (p, q) which is compactly supported in M̃ . Then
△h ϕ =

−g ki ∇δi ∇δk ϕ − g ki e(dxs )i(δk )(∇δs ∇δi − ∇δi ∇δs )ϕ + g ki Ghik ∇δh ϕ
³
´
+g ki Lhji|s + Lhjs|i + Ltji Lhst − Ltjs Lhit e(dxs )e(δy j )i(δk )i(∂˙h )ϕ
³
´
+g ki Lhik |s e(dxs )i(δh )ϕ + g ki Ltks|i − Lhis Lthk − Lhik Lths e(δy s )i(∂˙t )ϕ
−g ki Lhji Ltks e(δy j )e(δy s )i(∂˙h )i(∂˙t )ϕ.

Proof. By (4.1) and (4.10), we get
dh
d∗h

=
=

e(dxs )∇δs − Ltrs e(dxr )e(δy s )i(∂˙t ),
−g ki i(δk )∇δi + g ki Lhji e(δy j )i(δk )i(∂˙h ) + g ki Lhik i(δh ).

Thus
dh ◦ d∗h

=

−e(dxs )∇δs g ki i(δk )∇δi + e(dxs )∇δs g ki Lhji e(δy j )i(δk )i(∂˙h )
+e(dxs )∇δs g ki Lhik i(δh ) + g ki Ltrs e(dxr )e(δy s )i(∂˙t )i(δk )∇δi
−g ki Lh Lt e(dxr )e(δy s )i(∂˙t )e(δy j )i(δk )i(∂˙h )
ji

−g

ki

rs

Lhik Ltrs e(dxr )e(δy s )i(∂˙t )i(δh ).

Since the Cartan connection ∇ is horizontal metrical, it follows that
(4.12)

k
i
g ki|s = δs (g ki ) + g ti Γt;s
+ g kt Γt;s
= 0,

where | denotes the horizontal covariant derivative of ∇. Using Lemma 4.3 and (4.12),

176

Chunping Zhong

we obtain
dh ◦ d∗h

=

i
g kt Γt;s
e(dxs )i(δk )∇δi − g ki e(dxs )i(δk )∇δs ∇δi
+g ki P h e(dxs )e(δy j )i(δk )i(∂˙h ) + g ki Lh e(dxs )e(δy j )i(δk )i(∂˙h )∇δ
ji

ji|s

s

+g ki Lhik |s e(dxs )i(δh ) + g ki Lhik e(dxs )i(δh )∇δs
−g ki Ltrs e(dxr )e(δy s )i(δk )i(∂˙t )∇δi − g ki Lhji Ljrs e(dxr )e(δy s )i(δk )i(∂˙h )
−g ki Lh Lt e(dxr )e(δy s )e(δy j )i(δk )i(∂˙t )i(∂˙h )
ji

+g

ki

rs

Lhik Ltrs e(dxr )e(δy s )i(δh )i(∂˙t ).

On the other hand, it is easy to check that
d∗h ◦ dh

s
s
e(dxl )i(δk )∇δs − g ki ∇δi ∇δk + g ki e(dxs )i(δk )∇δi ∇δs
g ki Γk;i
∇δs − g ki Γl;i
+g ki Ltks|i e(δy s )i(∂˙t ) + g ki Ltrs|i e(dxr )e(δy s )i(δk )i(∂˙t )

=

+g ki Ltks e(δy s )i(∂˙t )∇δi + g ki Ltrs e(dxr )e(δy s )i(δk )i(∂˙t )∇δi
−g ki Lhji e(δy j )i(∂˙h )∇δk − g ki Lhji e(dxs )e(δy j )i(δk )i(∂˙h )∇δs
+g ki Lh Lt e(δy j )i(∂˙t ) − g ki Lh Lt e(δy j )e(δy s )i(∂˙h )i(∂˙t )
ji

+g

ki

kh

ji

ks

Lhji Ltrh e(dxr )e(δy j )i(δk )i(∂˙t )

+g ki Lhji Ltrs e(dxr )e(δy j )e(δy s )i(δk )i(∂˙h )i(∂˙t )
+g ki Lhik ∇δh − g ki Lhik e(dxs )i(δh )∇δs
−g ki Lhik Lths e(δy s )i(∂˙t ) − g ki Lhik Ltrs e(dxr )e(δy s )i(δh )i(∂˙t ),
where Ltrs|i denote the horizontal covariant derivative of Ltrs with respect to ∇. Now
sum dh ◦ d∗h and d∗h ◦ dh together and rearrange the resulted terms and indexes, we
obtain the expression of △h in Theorem 4.5.
¤
Remark 4.6. The horizontal Laplacian for functions f ∈ C ∞ (M̃ ) was first derived
in [4], and the horizontal and vertical Laplacians for 1-forms on the total space E of
Riemann vector bundle p : E → M was first derived in [8] under the assumption that
the fiber space F = p−1 (x), x ∈ M , is compact.
Note that if F is a Landsberg metric then Lijk = 0. Thus we have
Corollary 4.7. Let (M, F ) be a Landsberg manifold with the Cartan connection ∇,
and ϕ be a differential form of type (p, q) which is compactly supported in M̃ . Then
△h ϕ =

−g ki ∇δi ∇δk ϕ − g ki e(dxs )i(δk )(∇δs ∇δi − ∇δi ∇δs )ϕ + g ki Ghik ∇δh ϕ.

Corollary 4.8. Let (M, F ) be a real Finsler manifold with the Cartan connection ∇,
and ϕ be a horizontal differential form of type p which is compactly supported in M̃ .
Then
△h ϕ

=

−g ki ∇δi ∇δk ϕ − g ki e(dxs )i(δk )(∇δs ∇δi − ∇δi ∇δs )ϕ
+g ki Ghik ∇δh ϕ + g ki Lhik |s e(dxs )i(δh )ϕ.

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Laplacians on the tangent bundle of Finsler manifold

Theorem 4.9. Let (M, F ) be a real Finsler manifold with the Cartan connection ∇,
and ϕ be a differential form of type (p, q) which is compactly supported in M̃ . Then
³
´
1
h
∇∂˙h ϕ
△v ϕ = −g ki ∇∂˙i ∇∂˙k ϕ − g ki e(δy s )i(∂˙k ) ∇∂˙s ∇∂˙i − ∇∂˙i ∇∂˙s ϕ − g ki Rki
2
³
1 h´
h
e(δy s )i(∂˙h )ϕ
−g ki Cki
+ Rki
2
ks
h
i
k
t
k
t
k
−g hi 2Cjiks
− (Cji
Cts
− Cjs
Cti
) e(dxj )e(δy s )i(δk )i(∂˙h )ϕ
h
i
1 h a
a
h a
h a
+ Rki
−g ki Cskki
Csh + (Cki
Csh − Csi
Ckh ) e(dxs )i(δa )ϕ
2
t
h
−g ki Cji
Cst
e(dxj )e(dxs )i(δk )i(δh )ϕ.
Proof. By Lemma 4.1 and Lemma 4.4, we have
a
e(dxs )e(δy t )i(δa )
dv = e(δy s )∇∂˙s − Cst

(4.13)
and
(4.14)

³
1 h´ ˙
h
h
d∗v = −g ki i(∂˙k )∇∂˙i − g ki Cki
+ Rki
i(∂h ) + g ki Cji
e(dxj )i(δk )i(∂˙h ).
2

Since ∇ is v-metrical, it follows that the vertical covariant derivatives of g ki vanish
identically, i.e.,
k
i
g kiks = ∂˙s (g ki ) + g ti Cts
+ g kt Cts
≡ 0.
Using this fact and the properties of the operators i and e, we obtain
dv ◦ d∗v

k
= g ti Cst
e(δy s )i(∂˙k )∇∂˙i − g ki e(δy s )i(∂˙k )∇∂˙s ∇∂˙i
³
³
1 h´
1 h´
h
h
−g ki Cki
+ Rki
e(δy s )i(∂˙h )∇∂˙s
e(δy s )i(∂˙h ) − g ki Cki
+ Rki
2
2
ks
i
h
³
1 h´
t
h
h
k
h
− g ki Cjs
Cti
e(dxj )e(δy s )i(δk )i(∂˙h )
+ − g ki Cjiks
+ g ti Cjs
Cti
+ Rti
2
k
h
e(dxj )e(δy s )i(δk )i(∂˙h )∇∂˙i
−g ki Cji
e(dxj )e(δy s )i(δk )i(∂˙h )∇∂˙s + g hi Cjs
−g ki C a C h e(dxs )e(dxj )e(δy t )i(δa )i(δk )i(∂˙h ).
st

ji

By similar calculations, we have
d∗v ◦ dv

1
h
= − g ki Rki
∇∂˙h − g ki ∇∂˙i ∇∂˙k + g ki e(δy s )i(∂˙k )∇∂˙i ∇∂˙s
2
³
1 h´
i
h
−g kt Cst
e(δy s )i(∂˙k )∇∂˙i + g ki Cki
+ Rki
e(δy s )i(∂˙h )∇∂˙s
2
³
´
i
a
+ g at Cst
− g ki Csk
e(dxs )i(δa )∇∂˙i
³
h
1 h´ a i
h
a
h a
+ Rki
+ − g ki Cskki
+ g ki Csi
Ckh − g ki Cki
Csh e(dxs )i(δa )
2
³
h
i
1 h´ k
h
k
h k
+ Rti
+ − g hi Cjski
− g ti Cti
Cjs + g ti Cji
Cts e(dxj )e(δy s )i(δk )i(∂˙h )
2
t
h
k
−g ki Cji
Cst
e(dxj )e(dxs )i(δk )i(δh ) − g hi Cjs
e(dxj )e(δy s )i(δk )i(∂˙h )∇∂˙i
+g ki C h e(dxj )e(δy s )i(δk )i(∂˙h )∇ ˙ + g ki C h C a e(dxj )e(dxs )e(δy t )i(δk )i(δa )i(∂˙h ).
ji

∂s

ji

st

178

Chunping Zhong

h
k
Note that g ki Cji
= g hi Cji
and g kiks = 0, thus
h
t
h
k
h k
g ki Cjiks
+ g ki Cjs
Cti
+ g hi Cjski
− g ti Cji
Cts
³
´
´
³
t
k
t
k
k
k
Cts
− Cjs
Cti
.
− g hi Cji
+ Cjski
g hi Cjiks

=

k
k
Now it is easy to check that Cjiks
= Cjski
. Thus sum dv ◦ d∗v and d∗v ◦ dv together and
rearrange the resulted terms, we obtain △v in Theorem 4.9.
¤

Corollary 4.10. Let ∇ be the Cartan connection associated to a real Finsler manifold
(M, F ), and ϕ be a horizontal differential form of type p which is compactly supported
in M̃ . Then
△v ϕ

5

1
h
∇∂˙h ϕ
= −g ki ∇∂˙i ∇∂˙k ϕ − g ki Rki
2
³
1 h a´
a
h a
h a
−g ki Cskki
− Csi
Ckh + Cki
Csh + Rki
Csh e(dxs )i(δa )ϕ
2
t
h
−g ki Cji
Cst
e(dxj )e(dxs )i(δk )i(δh )ϕ.

The mixed-type operator

In this section, we shall derive the local expression of △mix = dmix ◦d∗mix +d∗mix ◦dmix .
Theorem 5.1. Let ϕ be a differential form of type (p, q) which is compactly supported
in M̃ . Then

(5.1)

1
j
s
Rli
gsj e(dxa )e(dxc )i(δk )i(δh )ϕ
△mix ϕ = g ki g lh Rac
4
1
t
s
+ g ki g lh Rhk
Rli
gsj e(δy j )i(∂˙t )ϕ
2
t
s
+ g ki g lh Rhc
Rli
gsj e(dxc )e(δy j )i(δk )i(∂˙t )ϕ.

Proof. On the one hand,
dmix ◦ d∗mix

=

1 ki lh j s
g g Rac Rli gsj e(dxa )e(dxc )i(δk )i(δh )
4
1
t
s
Rli
gsj e(dxa )e(dxc )e(δy j )i(δk )i(δh )i(∂˙t ).
− g ki g lh Rac
4

On the other hand,
d∗mix ◦ dmix

=

1 ki lh t s
g g Rac Rli gsj e(δy j )i(δk )i(δh )e(dxa )e(dxc )i(∂˙t ).
4

Using the relationship i(δh )e(dxa ) = δha − e(dxa )i(δh ) repeatedly, we get
i(δk )i(δh )e(dxa )e(dxc ) =

δha δkc − δha e(dxc )i(δk ) − δka δhc + δka e(dxc )i(δh ) + δhc e(dxa )i(δk )
−δkc e(dxa )i(δh ) + e(dxa )e(dxc )i(δk )i(δh ).

Laplacians on the tangent bundle of Finsler manifold

179

Thus
d∗mix ◦ dmix

=

1 ki lh t
t
s
g g (Rhk − Rkh
)Rli
gsj e(δy j )i(∂˙t )
4
1
t
s
+ g ki g lh Rhc
Rli
gsj e(dxc )e(δy j )i(δk )i(∂˙t )
4
1
t
s
− g ki g lh Rkc
Rli
gsj e(dxc )e(δy j )i(δh )i(∂˙t )
4
1
t
s
Rli
gsj e(dxa )e(δy j )i(δk )i(∂˙t )
− g ki g lh Rah
4
1
t
s
+ g ki g lh Rak
Rli
gsj e(dxa )e(δy j )i(δh )i(∂˙t )
4
1
t
s
+ g ki g lh Rac
Rli
gsj e(dxa )e(dxc )e(δy j )i(δk )i(δh )i(∂˙t ).
4

i
i
Sum dmix ◦ d∗mix and d∗mix ◦ dmix together, and use the fact Rjk
= −Rkj
, we get (5.1).
¤

Remark 5.2. It follows from Theorem 4.5 and Theorem 4.9 that the local expressions
of △h and △v depend on the choice of Finsler connection associated to (M, F ), while
the expression of △mix is independent of the choice of Finsler connections associated
to (M, F ).
Corollary 5.3. Let ϕ be a horizontal form of type p which is compactly supported in
M̃ . Then
△mix ϕ

=

1 ki lh j s
g g Rac Rli gsj e(dxa )e(dxc )i(δk )i(δh )ϕ.
4

Especially, for every horizontal 1-form ϕ = ϕi dxi which is compactly supported in M̃ ,
we have △mix ϕ ≡ 0.

6

Some properties of the type preserving operators

Let △ be the Hodge-Laplace operator of g̃ on M̃ . It is clear that △, △h , △v , △mix
satisfy
(6.1)

(△ϕ, ψ) = (△h ϕ, ψ) + (△v ϕ, ψ) + (△mix ϕ, ψ),

which implies that △ϕ = 0 if and only if
(6.2)

△h ϕ = 0, △v ϕ = 0, △mix ϕ = 0, ∀ϕ ∈ ∧p,q
c (M̃ ).

This is essentially arisen from the type-preserving property of △h , △v and △mix .
Especially if f ∈ Cc∞ (M̃ ), then
(6.3)

0 ≤ (df, df ) = (△f, f ) = (△h f, f ) + (△v f, f ),

which implies that △f = 0 if and only if △h f = 0 and △v f = 0.

180

Chunping Zhong

Proposition 6.1. Let λ, λh and λv be constants such that
△f = λf,

△h f = λh f,

△v f = λv f,

Then λ ≥ 0, λh ≥ 0, λv ≥ 0 and
(6.4)

λ = λh + λv .

Proof. It is clear.

¤

Theorem 6.2. Let f be a scalar field on M̃ which is not constant and satisfies
△h f = λh f, △v f = λv f with λh , λv being a constant such that λ2h + λ2v 6= 0. Then
the constant λ =: λh + λv must be positive.
Proof. It is easy to check that
1
△h f 2 = f △h f − g ij (∇δi f )(∇δj f ),
2

1
△v f 2 = f △v f − g ij (∇∂˙i f )(∇∂˙j f ).
2

Since f is compactly supported in M̃ , thus
Z h
Z
i
1
△f 2 dV =
(6.5)
0=
f △f − g ij (∇δi f )(∇δj f ) − g ij (∇∂˙i f )(∇∂˙j f ) dV.
M̃
M̃ 2
Substituting the identity △f = △h f + △v f in (6.5), we get
Z h
i
λf 2 − g ij (∇δi f )(∇δj f ) − g ij (∇∂˙i f )(∇∂˙j f ) dV = 0.
M̃

If on the contrary λ < 0, then the above equality implies that ∇δi f = ∇∂˙i f ≡ 0, that
is, f is a constant. This is a contradiction to the assumption.
¤
Theorem 6.3. There exists no horizontal 1-form ϕ = ϕi dxi with compact support in
M̃ which satisfies relations
´
³
³
´
1 h
(6.6)
g ki ϕl|k|i − Ghki ϕl|h + g ki ϕlkkki + Rki
ϕlkh = Ttl ϕt
2
and
Ttl ϕt ϕl ≥ 0

(6.7)
unless we have
(6.8)

ϕl|k = 0, ϕlkk = 0

and then automatically Ttl ϕt ϕl = 0.
Proof. Denote kϕk2 := g lt ϕl ϕt and ϕl := g lt ϕt . Then it is easy to check that
´
³
△h kϕk2 = −2g ki ϕl|k|i − Ghki ϕl|h ϕl − 2g ki g lt ϕl|i ϕt|k ,
³
´
1 h
△v kϕk2 = −2g ki ϕlkkki + Rki
ϕlkh ϕl − 2g ki g lt ϕlkk ϕtki .
2
Thus if condition (6.6) and (6.7) are satisfied, we have
1
− △kϕk2 = Ttl ϕt ϕl + g ki g lt (ϕl|i ϕt|k + ϕlkk ϕtki ) ≥ 0,
2
from which we get (6.8).

¤

Laplacians on the tangent bundle of Finsler manifold

181

Remark 6.4. Note that a function f ∈ C ∞ (M̃ ) which is both horizontal and vertical
parallel with respect to the Cartan connection ∇ is necessary a constant. It follows
from Theorem 6.3 that the only exceptions are vector fields which are both horizontal
and vertical parallel, and there are no such vector fields other than zero if the quadratic
form Ttl ϕt ϕl is positive definite.
Acknowledgment. This work was supported by Program for New Century Excellent Talents in Fujian Province University, and the Natural Science Foundation of
China(Grant No. 10971170, 10601040).

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[8] O. Munteanu, Weitzenböck formulas for horizontal and vertical Laplacians, Houston J. Math., 29, 4 (2003), 889-900.
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Author’s address:
Chunping Zhong
School of Mathematical Sciences,
Xiamen University,
Xiamen 361005, P.R. China,
E-mail: zcp@xmu.edu.cn