Adiabatic limit and connections in Finsler Geometry

arXiv:1207.1552v3 [math.DG] 30 Oct 2012

ADIABATIC LIMIT AND CONNECTIONS IN FINSLER GEOMETRY HUITAO FENG1 AND MING LI2

Abstract. In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang [7], the Cartan connection can also be obtained through an adiabatic limit process. Furthermore, a Chern-Simons type form is defined and its conformal properties are discussed. Keywords: Bott connection, Chern connection, Cartan connection, adiabatic limit, Chern-Simons type form

Introduction In Finsler geometry, the Chern connection and the Cartan connection are two basic connections which have remarkable properties. Let (M, F ) be a Finsler manifold. Let π : SM → M be the projective sphere bundle of M. Then the Finsler structure F on M defines naturally an Euclidean structure on the pull-back vector bundle π ∗ T M → SM and a Sasaki-type Riemannian metric on SM. The Chern and Cartan connections are connections on π ∗ T M and defined from different geometric reasons. On the other hand, the Finsler structure F gives rise to a natural splitting of T (SM). One part is the vertical tangent bundle V (SM) formed by the tangent vectors of the (vertical) projective spheres, which is an integrable subbundle of T (SM). Another part is the horizontal tangent bundle H(SM), which is defined as the orthogonal complement of V (SM) in T (SM) with respect to the Sasaki-type Riemannian metric on SM. It is well-known that H(SM) with its restriction metric is isometric to π ∗ T M. In this paper, we consider SM as a foliation foliated by projective spheres. So the well-known Bott connection in foliation theory is now a connection on H(SM). We will prove that the Bott connection is the Chern connection under the identification of H(SM) and π ∗ T M. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection. These also partially answer a question of M. Abate and G. Patrizio (cf. [1, p.29]). Following Liu-Zhang [7], the relations between the Chern connection, the Cartan connection and the Levi-Civita connection 1

Partially supported by NSFC (Grant No. 10921061), Fok Ying Tong Education Foundation, Chongqing NSF and the Scientific research project of CQUT. 2 Partially supported by Chongqing NSF (Grant No. cstc2011jjA00026) and the Scientific research project of CQUT (Grant No. 01-60-37). 1

2

HUITAO FENG AND MING LI

associated to the Sasaki-type Riemannian metric are also established through an adiabatic limit process. We then consider a special Chern-Simons transgressed term of the Chern and Cartan connections. In the case of dimension 2, the explicit formula of this term is given. Inspired by this formula, we define a Chern-Simons type form of (M, F ), which is a non-Riemannian geometric invariant of the Finsler manifold. Some conformal properties of this form are also discussed. This paper is organized as follows. In Section 1, we give a review of some basic facts in Finsler geometry. In Section 2, we study the relations between the Bott connection, the Chern connection and the Cartan connection for a Finsler manifold. In Section 3, we define a Chern-Simons type form of a Finsler manifold and discuss its conformal properties. Acknowledgements. The first author would like to thank Professor Weiping Zhang for his consistent support and encouragement. The authors thank Professors Kefeng Liu and Weiping Zhang for their many helpful suggestions in preparing this paper. 1. Finsler manifolds and related structures In this section we give a review of some basic facts in Finsler geometry which will be used in this paper. Let M be an n dimensional smooth manifold and π : T M → M the tangent bundle of M. Let (U; x = (x1 , x2 , . . . , xn )) be a local coordinate system on an open subset U of M. Then by the standard procedure one gets a local coordinate system (x, y) = (x1 , . . . , xn , y 1, . . . , y n ) on π −1 (U). Set T M0 = T M \ 0, where 0 denotes the zero section of T M. Then (x, y) with y 6= 0 is a local coordinate system on T M0 . Definition 1. A Finsler structure on M is a smooth function F : T M0 → R+ , which satisfies the following conditions: (i) F (x, λy) = λF (x, y), ∀(x, y) ∈ T M0 , and λ ∈ R+ ; (ii) The n × n Hessian matrix 1 2 (gij ) = F yi yj 2 is positive-definite at every point of T M0 . A manifold M with a Finsler structure F is called a Finsler manifold, and denoted by (M, F ). In this paper, lower case Latin indices will run from 1 to n and lower case Greek indices will run from 1 to n−1. We also adopt the summation convention of Einstein. Let (M, F ) be an n-dimensional Finsler manifold. Set 2 1 ij 2 k i (1.1) F y j xk y − F xj , G = g 4 (1.2)

δ ∂ ∂Gj ∂ = − , δxi ∂xi ∂y i ∂y j

δ ∂ = F i, i δy ∂y

ADIABATIC LIMIT AND CONNECTIONS IN FINSLER GEOMETRY

3

where (g ij ) = (gij )−1 . Clearly, the vectors δ δ δ δ δ δ (1.3) , , . . ., n, 1, 2, . . ., n δx1 δx2 δx δy δy δy

form a local tangent frame of T M0 . For another local coordinate system (U; x˜) on M, a routine computation shows that

δ ∂ x˜j δ δ ∂ x˜j δ (1.4) = , = . δxi ∂xi δ˜ xj δy i ∂xi δ˜ yj Now by (1.4), one gets a well-defined linear map J : T (T M0 ) → T (T M0 ) δ δ δ δ (1.5) = i, J = − i, J i i δx δy δy δx which is in fact an almost complex structure on T M0 . Let 1 (1.6) δx , δx2 , . . . , δxn , δy 1 , δy 2, . . . , δy n be the dual frame of (1.3). One has (1.7)

i

i

δx = dx ,

1 δy = F i

and J ∗ (δxi ) = −δy i ,

(1.8)

∂Gi j i dy + j dx , ∂y J ∗ (δy i ) = δxi ,

where J ∗ denotes the dual map of J. Let π : SM = T M0 /R+ → M denote the projective sphere bundle. Now the fundamental tensor g = gij dxi ⊗ dxj defines an Euclidean metric on the pull back bundle π ∗ T M over SM. Note that π ∗ T M admits a distinguished global section l : SM → π ∗ T M, which is defined by y (1.9) . l(x, [y]) = x, [y], F (x, y)

For any local orthonormal frame {e1 , . . . , en } of (π ∗ T M, g) with en = l, let {ω 1 , · · · , ω n } be the dual frame. Clearly, ω i ’s can be also viewed naturally as (local) one forms on SM as well as on T M0 . Here ω n , the so called Hilbert form, is a globally defined one form and ω n = Fyi δxi . Set ω n+i = J ∗ (ω i ),

(1.10)

i = 1, 2, . . . , n.

The one forms ω 1 , ω 2, . . . , ω 2n−1 and ω 2n = −Fyi δy i give rise to a local coframe of T M0 . Moreover, one verifies easily that the forms ω n+α , α = 1, 2, . . . , n − 1, are actually the one forms on SM (cf. [6, p.269]) and the set (1.11) θ = ω 1 , . . . , ω n , ω n+1, . . . , ω 2n−1 forms a local coframe of SM. By using the local coframe (1.11), the tensor (1.12)

g

T (SM )

=

n X i=1

i

i

ω ⊗ω +

n−1 X α=1

ω n+α ⊗ ω n+α

4


gives a well-defined Riemannian metric on SM, which is called the Sasaki-type Riemannian metric on SM. Moreover, the fundamental tensor g can be written as n n n X X X i j i j (1.13) g= gij dx ⊗ dx = gij δx ⊗ δx = ω i ⊗ ω i on SM. i,j=1

i,j=1

i=1

As mentioned in the introduction of this paper, the vertical and horizontal subbundles V (SM) and H(SM) of T (SM) are orthogonal to each other with respect to g T (SM ). Let {e1 , . . . , en , en+1 , . . . , e2n−1 } denote the dual frame of θ. Note that (1.14)

{e1 , e2 , . . . , en−1 , en }

is a local orthonormal frame of H(SM). Remark 1. (π ∗ T M, g) can be identified with H(SM) with the restricting metric of g T (SM ) as Euclidean bundles. In fact, this identification is given by identifying ∂x∂ i i with δxδ i and so ei with ei . In particular, the distinguished section l = en = yF ∂x∂ i in i (1.9) turns out to be the Reeb vector field G = en = yF δxδ i of (M, F ) on SM. Write that ω j = vij δxi ,

(1.15)

and so ω n+α = J ∗ (viα δxi ) = −viα δy i .

Then one has ei = uji

(1.16)

δ δxj

and en+α = −ujα

δ , δy j

where (uij ) = (vij )−1 . Here also note that vin = Fyi and uin = gets easily that n−1 X

(1.17)

viα vjα = F Fyi yj ,

yi . F

By Definition 1, one

F Fyi g ij = y j .

α=1

The following lemma gives an explicit expression of the exterior derivative of the Hilbert form ω n with respect to the local coframe (1.11). This formula is usually obtained as one of the structure equations of the Chern connection in Finsler geometry. Lemma 1. The exterior derivative of Hilbert form is given by n

(1.18)

dω =

n−1 X

ω α ∧ ω n+α.

α=1

Proof. Note that

dω n = d(Fyi δxi ) = Fyi xj δxj ∧ δxi + Fyi yj dy j ∧ δxi ∂Gk j i = F Fyi yj δy ∧ δx + Fyi xj − Fyi yk δxj ∧ δxi j ∂y δF ∂ 2 Gk j i = F Fyi yj δy j ∧ δxi + F k δxj ∧ δxi . δx ∧ δx + δxj yi ∂y j ∂y i y


5

Pn−1 α n+α ∂ 2 Gk j ω ∧ω . Clearly, ∂y By (1.15) and (1.17), the term F Fyi yj δy j ∧δxi = α=1 j ∂y i Fy k δx ∧ δxi = 0. Now the lemma follows from the following result (cf. [3, p.36]), ∂Gk δF = Fxj − F k = 0. δxj ∂y j y

(1.19)

The following lemma is actually obtained by Mo in [9]. We will give it a direct proof without using any concepts of connections. Lemma 2 (Mo, [9]). The Lie derivative of the fundamental tensor g along the Reeb vector field G (cf. Remark 1.) is given by LG g = −

(1.20)

n−1 X α=1

Proof. Firstly one has

ω α ⊗ ω n+α + ω n+α ⊗ ω α .

yk δ 1 yk ∂ ∂Gl ∂ 1 2 2 2 G(gij ) = [F ]yi yj = [F ]yi yj − k l [F ]yi yj F δxk 2 2 F ∂xk ∂y ∂y 2 l l k 1y δ[F ] ∂G ∂G 1 = glj i + gli j + k 2F δx F ∂y ∂y yi yj ∂Gl ∂Gl 1 glj i + gli j . = F ∂y ∂y

Then by (1.13) and Cartan homotopy formula (cf. [11, p.30]), one has LG g = LG (gij dxi ⊗ dxj ) = G(gij )dxi ⊗ dxj + gij LG (dxi ) ⊗ dxj + gij dxi ⊗ LG (dxj ) 1 ∂Gl ∂Gl = glj i + gli j dxi ⊗ dxj F ∂y ∂y i yi dy j yj dy ⊗ dxj − gij d log F ⊗ dxj + gij dxi ⊗ − gij dxi ⊗ d log F + gij F F F F i j i j i j = gij δy ⊗ dx + gij dx ⊗ δy − d log F ⊗ Fyj dx − Fyi dx ⊗ d log F n n X X i n+i =− ω ⊗ω − ω n+i ⊗ ω i − d log F ⊗ ω n − ω n ⊗ d log F =−

i=1

i=1

n−1 X

ω α ⊗ ω n+α + ω n+α ⊗ ω α .

α=1

The last equation comes from that ω 2n = −d log F , a direct corollary of (1.19). Remark 2. We denote the Hilbert form as ω = ω n . By Lemma 1, one has that ω ∧ (dω)n−1 6= 0. So ω is a contact form of SM.

6


2. The relations of some connections related to a Finsler manifold In this section we will use the same notations as in Section 1. Note that there exists a natural foliation structure on the Riemannian manifold (SM, g T (SM ) ), which is foliated by the vertical bundle V (SM). Following Liu-Zhang [7] and Zhang [11, Sect. 1.7], set (2.1)

F = V (SM),

F ⊥ = H(SM).

Let ∇T (SM ) be the Levi-Civita connection on T (SM) associated to the Sasaki-type Riemannian metric g T (SM ) on SM. Let p, p⊥ denote the orthogonal projections from T (SM) to F , F ⊥ respectively. Set ∇F = p∇T (SM ) p,

(2.2)

⊥

∇F = p⊥ ∇T (SM ) p⊥ .

⊥

⊥

Let g F , g F be the restriction of g T (SM ) on F , F ⊥ respectively. Then ∇F , ∇F are metric-preserving connections of F , F ⊥ respectively. e F ⊥ on F ⊥ is determined by the following definition Now the Bott connection ∇ Definition 2 (cf. [7], [11, Sect. 1.7]). For any X ∈ Γ(T (SM)), U ∈ Γ(F ⊥ ), e F ⊥ U = p⊥ [X, U]; (i) If X ∈ Γ(F ), set ∇ X e F ⊥ U = ∇F ⊥ U. (ii) If X ∈ Γ(F ⊥ ), set ∇ X X

e F ⊥ is not a metric-preserving connection of g F ⊥ . In general, the Bott connection ∇ e F ⊥ ,∗ of the Bott connection as follows, One defines the dual connection ∇ e F ⊥ U, V i + hU, ∇ e F ⊥ ,∗ V i, dhU, V i = h∇

where U, V ∈ Γ(F ⊥ ). Following Bismut-Zhang [4, p.62] and Liu-Zhang [7], set (2.3)

e F ⊥ ,∗ − ∇ e F⊥ 2H = ∇

b F⊥ = ∇ e F ⊥ + H. and ∇

b F ⊥ is the symmetrization of the Bott connection It is known that the connection ∇ ⊥ with respect to the metric g F on F ⊥ and so a metric-preserving connection on F ⊥ . Some basic properties of the Ω1 (SM)-valued endomorphism H are also established in [4, p.62] and [7]. Lemma 3 ([4, p.62], [7]). For any U, V ∈ Γ(F ⊥ ), one has that (1) hHU, V i=hU, HV i, (2) H(U) = 0, ⊥ e F ⊥ gF ⊥ . (3) H = 21 (g F )−1 ∇

Write H = (Hij ) under the local frame (1.14). As a corollary of Lemma 3, one has that Hij = Hji and Hij = Hijγ ω n+γ for some functions Hijγ . Lemma 4. Set Aijk = 14 F [F 2 ]yi yj yk . With respect to (1.16), one has (2.4)

Hijγ = −Apqk upi uqj ukγ .

Moreover, Hijγ = 0 if i = n or j = n.


7

Proof. For any X ∈ Γ(F ) and U, V ∈ Γ(F ⊥ ), one gets easily that ⊥

h2H(X)U, V i = (LX g F )(U, V ). So by (1.13), (1.16) and (2.4), 1 1 ⊥ Hijγ = hH(en+γ )ei , ej i = (Len+γ g F )(ei , ej ) = (Len+γ g)(ei , ej ) 2 2 1 1 ∂g pq = (en+γ gpq )dxp ⊗ dxq (ei , ej ) = − F ukγ k dxp ⊗ dxq (ei , ej ) 2 2 ∂y 1 = − F [F 2 ]yp yq yk upi uqj ukγ = −Apqk upi uqj ukγ . 4 By the Euler lemma, it is clear that Hijγ = 0 if i = n or j = n.

Remark 3. Traditionally, the Cartan tensor is defined as A = Aijk dxi ⊗ dxj ⊗ dxk , and the Cartan form is that I = g ij Aijk dxk := Ak dxk (cf. [8, p.11-12]). From this reason, we call H the Cartan endomorphism, and still call the one form η = tr[H] ∈ Ω1 (SM) the Cartan form for a Finsler manifold (M, F ). Let ω = (ωji ) be the connection matrix of the Bott connection with respect to the orthnormal frame (1.14), i.e., e F ⊥ ei = ω j ej . ∇ i

(2.5)

Theorem 1. The connection matrix ω = (ωji ) in (2.5) of the Bott connection is the unique solution of the following structure equations, ( dϑ = ϑ ∧ ω, (2.6) ω + ω t = −2H, where ϑ = (ω 1, . . . , ω n ). Proof. For any X, Y ∈ Γ(T (SM), (dω i − ω j ∧ ωji )(X, Y )

=X(ω i (Y )) − Y (ω i (X)) − ω i ([X, Y ]) − ω j (X)ωji (Y ) − ω j (Y )ωji (X) .

Now for any X, Y ∈ Γ(F ), and U, V ∈ Γ(F ⊥ ), one has

(dω i − ω j ∧ ωji )(X, Y ) = −ω i ([X, Y ]) = 0, (dω i − ω j ∧ ωji )(X, U) = X(ω i (U)) + ω j (U)ωji (X) − ω i ([X, U]) ⊥ e F U − [X, U] = 0, =ω i ∇ X

and

(dω i − ω j ∧ ωji )(U, V )

= U(ω i (V )) + ω j (V )ωji (U) − V (ω i (U)) + ω j (U)ωji (V ) − ω i ([U, V ]) ⊥ eF V − ∇ e F ⊥ U − [U, V ] = ω i ∇T (SM ) V − ∇T (SM ) U − [U, V ] = 0. =ω i ∇ U

V

U

V

8


Hence the Bott connection matrix ω satisfies the first equation of (2.6). The second equation of (2.6) comes directly from the definition of H. e = (e To prove the uniqueness, let ω ωji ) be another solution of (2.6). One has ω j ∧ (e ωji − ωji ) = 0.

It deduces that ω eji − ωji = aijk ω k ,

with aijk = aikj .

From the second equation of (2.6), one has that 0 = (ωji + ωij ) − (e ωji + ω eij ) = (aijk + ajik )ω k ,

and so aijk + ajik = 0. Thus

(aijk + ajik ) + (akij + aikj ) − (ajki + akji ) = 2aijk = 0. So we conclude that ω eji − ωji = 0.

Corollary 1. The connection forms of the Bott connection in (2.5) satisfy ωαn = −ωnα = ω n+α ,

and

ωnn = 0.

Proof. The formula ωnn = 0 comes directly from Lemma 4. By Lemma 1, the connection forms ωαn can be written as (2.7)

ωαn = ω n+α + cαβ ω β

with cαβ = cβα .

The second equation of (2.6) and Lemma 4 imply that (2.8)

ωnα = −ωαn = −ω n+α − cαβ ω β .

By (2.8) and the first equation of (2.6), one has that dω α = ω β ∧ ωβα + ω n ∧ ωnα = ω β ∧ (ωβα + cαβ ω n ) + ω n ∧ (−ω n+α). e = (e Set ω eβα = ωβα + cαβ ω n , ω eαn = −e ωnα = ω n+α and ω enn = 0. Clearly, ω ωji ) satisfies the first equation of (2.6). Moreover, eβα = 2cαβ ω n − 2Hαβγ ω n+γ . ω eβα + ω


9

Note that by Cartan homotopy formula, one has Len g = Len

n X

i

i

ω ⊗ω =

i=1

=

n X

n X

(Len ω i ) ⊗ ω i + ω i ⊗ (Len ω i )

i=1

i

i

i

i

(ien dω ) ⊗ ω + ω ⊗ (ien dω ) =

i=1

=

n−1 X

α,β=1

=−

n−1 X

((ien dω α) ⊗ ω α + ω α ⊗ (ien dω α))

α=1

enα ) enα ) ⊗ ω α + ω α ⊗ ien (ω n ∧ ω eβα ) + ien (ω n ∧ ω eβα ) ⊗ ω α + ω α ⊗ ien (ω β ∧ ω ien (ω β ∧ ω

n−1 X

(e ωβα (en ) α,β=1

+

ω eαβ (en ))ω α

n−1 X ⊗ω − (ω α ⊗ ω n+α + ω n+α ⊗ ω α ). β

α=1

Comparing with Lemma 2, we conclude that 2cαβ = (e ωαβ + ω eβα )(en ) = 0.

Now by (2.7), the corollary is proved.

Remark 4. Noticed that the Chern connection is defined by the structure equations (2.6) (cf. [3, p.38], [6, p.282], [8, p.23-33]), so the Bott connection in our case is exactly the Chern connection. In this case, we partially answer a question of M. Abate and G. Patrizio (cf. [1, p.29]). Moreover, under the orthnormal frame (1.14), the b F ⊥ of the Bott connection has the connection matrix symmetrization ∇ b = ω + H. ω

(2.9)

In [3, p.39], an expression of the Cartan connection is given in the local coordinate system on SM. One can check easily that these two expressions are differ from a gauge b F ⊥ turns out to be the Cartan connection in transformation of the connection. So ∇ Finsler geometry. Now we consider the rescaled metrics on SM with ǫ > 0,

(2.10)

g T (SM ),ǫ =

n n−1 X 1 X i i ω ⊗ ω + ω n+α ⊗ ω n+α . ǫ2 i=1 α=1 ⊥

Let ∇T (SM ),ǫ be the Levi-Civita connection of g T (SM ),ǫ and ∇F ,ǫ = p⊥ ∇T (SM ),ǫ p⊥ . b F ⊥ now Following Liu-Zhang [7] and Zhang [11, Sect. 1.7], the Cartan connection ∇ can also be obtained through the adiabatic limit technique, i.e., Proposition 1. Let ∇F

⊥ ,ǫ

= p⊥ ∇T (SM ),ǫ p⊥ , then lim ∇F ǫ→0

⊥ ,ǫ

b F⊥. =∇

Furthermore, by using the technique of the adiabatic limit, we can prove the following property of the Cartan endomorphism H.

10


Proposition 2. Let (M, F ) be a Finsler manifold. For any σ ∈ π ∗ C ∞ (M), let ¯ be the associated Cartan endomorphism, then g¯T (SM ) = e2σ g T (SM ) and H ¯ = H. H e F ⊥ and ∇ b F ⊥ be the Bott connection and its symmetrization correspondProof. Let ∇ ing g¯T (SM ) = e2σ g T (SM ) , respectively. Then the corresponding Cartan endomorphism ¯ is H ¯ =∇ b F⊥ − ∇ e F⊥. H Consider the rescaled coformal metrics 1 ⊥ g¯T (SM ),ǫ = 2 e2σ g F ⊕ e2σ g F ǫ

¯ = H(U) = 0 for and the projection connections ∇F ⊥ ,ǫ on F ⊥,ǫ . It is clear that H(U) ⊥ ⊥ any U ∈ Γ(F ). For any X ∈ Γ(F ), U, V ∈ Γ(F ), we have b F U, V i − h[X, U], V i ¯ hH(X)U, V i = h∇ X ⊥

⊥ ,ǫ

F = limh∇X ǫ→0

U, V i − h[X, U], V i

1 = lim e−2σ ǫ2 {XhU, V iσ,ǫ + UhX, V iσ,ǫ − V hX, Uiσ,ǫ ǫ→0 2 +h[X, U], V iσ,ǫ − h[X, V ], Uiσ,ǫ − h[U, V ], Xiσ,ǫ } − h[X, U], V i 1 = lim {XhU, V i + 2X(σ)hU, V i ǫ→0 2 +h[X, U], V i − h[X, V ], Ui − ǫ2 h[U, V ], Xi − h[X, U], V i 1 = {XhU, V i − h[X, U], V i − h[X, V ], Ui} 2 = hH(X)U, V i. 3. Geometric classes of Finsler manifolds Let (M, F ) be an oriented and closed Finsler manifold of dimension n. As in the e F ⊥ and ∇ b F ⊥ denote the Chern connection and the Cartan previous section, let ∇ connection on F ⊥ = H(SM), respectively. ⊥ ⊥ Let ∇F defined by t , t ∈ [0, 1], be a family of connections on F ⊥

⊥

⊥ e F ⊥ + t∇ b F⊥ = ∇ e F ⊥ + tH. ∇F = (1 − t)∇ t ⊥

2 F Let RtF = (∇F t ) be the curvature of ∇t . The term Z 1 h i ⊥ (3.1) −n tr H(RtF )n−1 dt 0

h i ⊥ appears naturally in the transgression formula associated to tr (RtF )n (cf. [11, p.16]).


11

⊥

With respect to (1.14), the curvature two forms of R0F are Ωij = dωji − ωjk ∧ ωki . By the first equation of (2.6) (also the lemma 1.14 in [11]), one can write Ωij as i i Ωij = Rjkl ω k ∧ ω l + Pjkγ ω k ∧ ω n+γ ,

(3.2)

i i where Rjkl and Pjkγ are some functions on SM. In the following we will compute the term (3.1) for a Finsler surface.

Theorem 2. Let (M, F ) be an oriented and closed Finsler surface. The term (3.1) is given by Z 1 h i F⊥ (3.3) −2 tr HRt dt = η ∧ dη, 0

3

where η = tr[H] = H111 ω is the Cartan form of (M, F ) (cf. Remark 3). Proof. Firstly one has that Z 1 h Z i F⊥ tr HRt dt =

i h h ⊥ i ⊥ 2 3 dt tr HR0F + tH ∇F , H + t H 0 0 1 h F⊥ i 1 3 F⊥ = tr HR0 + H ∇0 , H + H . 2 3

0

1

In the case of dim M = 2, by Corollary 1 and (3.2), one has

1 1 dω 3 = −R212 ω 1 ∧ ω 2 − P211 ω 1 ∧ ω 3.

With respect to the local frame (1.14), one gets 1 1 H111 ω 3 0 Ω1 ∗ H111 R212 ω1 ∧ ω2 + · · · ∗ F⊥ H= , R0 = = . 0 0 ∗ ∗ ∗ ∗ Thus

⊥ HR0F =

and

1 (H111 )2 R212 ω1 ∧ ω2 ∧ ω3 0 0 0

Z

0

On the other hand,

1

,

⊥ H[∇F 0 , H] =

1 −(H111 )2 R212 ω1 ∧ ω2 ∧ ω3 0 0 0

h i 1 ⊥ 1 tr −HRtF dt = (H111 )2 R212 ω1 ∧ ω2 ∧ ω3. 2

1 η ∧ dη = H111 ω 3 ∧ d(H111 ω 3 ) = −(H111 )2 R212 ω1 ∧ ω2 ∧ ω3.

So Theorem 2 follows.

Remark 5. In [10], Szabó proved that any two dimensional Berwald manifold is either locally Minkowskian or Riemannian. So the term (3.1) is identically zero for any two dimensional Berwald manifold. On the other hand, in [5], Bryant constructed a family 1 of two dimensional non-Riemannian Finsler manifolds with R212 = 1. From Theorem 2, the cohomology class associated to the term (3.1) of these Finsler manifolds are not zero. Motivated by Theorem 2 and Remark 5, we make the following definition.

,

12


Definition 3. For a closed and oriented Finsler manifold (M, F ) of dimension n, the top form η ∧ (dη)n−1 on SM is called the Chern-Simons type form of (M, F ). The corresponding class 2n−1 [η ∧ (dη)n−1] ∈ HdR (SM)

is called the Chern-Simons type secondary class of (M, F ). When (dη)k = 0 for some k ≥ 1, one gets a closed form η ∧ (dη)k−1 and so a class 2k−1 [η ∧ (dη)k−1] ∈ HdR (SM). It would be interesting to explore the properties of the Finsler manifolds with (dη)k = 0 and [η ∧ (dη)k−1] 6= 0. Note that the form η ∧ (dη)n−1 is unchanged about the conformal metrics in Proposition 2. In the following proposition, a condition on conformal Finsler metrics is given which leaves η unchanged. Proposition 3. Let (M, F ) be a Finsler manifold. Let F¯ = eσ F be a conformal deformation of F , where σ ∈ π ∗ C ∞ (M). Let η and η¯ be the Cartan forms of (M, F ) and (M, F¯ ), respectively. Then η¯ = η if and only if σ satisfies G(σ)I + A(I∗, dσ ∗ ) = 0

(3.4)

and hI∗ , dσ ∗ i = 0,

i

where G = yF δxδ i is the Reeb vector field on SM; A is the Cartan tensor and I is the usual Cartan form (cf. Remark 3); I∗ , dσ ∗ are the dual vector fields of I, dσ with respect to the metric g T (SM ) , respectively. i

Proof. By (1.1), one has G = Gi + σxk y k y i − 12 F 2 σxk g ki . Furthermore, i

∂Gi ∂G = + σxj y i + σxk y k δji − F Fyj σxk g ki + F Apqj g ip g qk σxk , ∂y j ∂y j and

! i ∂G j dy + j dx ∂y i ∂G i j k i ki ip qk j −σ 1 dy + + σxj y + σxk y δj − F Fyj σxk g + F Apqj g g σxk dx =e F ∂y j 1 σxj y i + σxk y k δji − F Fyj σxk g ki + F Apqj g ip g qk σxk δxj . = e−σ δy i + e−σ F

1 δy i = ¯ F

i

Corresponding to F¯ , one has that ω i = eσ ω i and ω n+γ = −eσ vji δy j . Now,

1 γ v (σ k y j + σxl y l δkj − F Fyk σxl g lj + F Apqk g jpg ql σxl )δxk F j x 1 + σxl y l vkγ δxk − g lj vjγ σxl Fyk δxk + vjγ Apqk g jp g ql σxl δxk F 1 + σxl y l ω γ + vjγ Apqk g jpg ql σxl δxk − g lj vjγ σxl ω n . F

−ω n+γ = vjγ δy j + = −ω n+γ = −ω n+γ


13

On the other hand, one sees easily from (2.4) that functions Hαβγ are unchanged under the above conformal deformations. Finally we obtain ¯ iiγ ω n+γ η¯ = H = Hiiγ ω n+γ −

1 σxl y l Hiiγ ω γ − Hiiγ vjγ Apqk g jpg ql σxl δxk + Hiiγ vjγ g lj σxl ω n F

yl σ l I + Aj g jp Apqk g ql σxl δxk − Aj g jl σxl ω n F x = η + G(σ)I + A(I∗ , dσ ∗ ) − hI∗, dσ ∗ iω n .

=η+

By the above proposition, the Chern-Simons type form η ∧ (dη)n−1 is a conformal invariant when the conformal factor σ satisfies (3.4). It should be noted that the second equation in (3.4) also appears as the conformal invariance condition of the so called S-curvature (cf. [2, p.231]). References [1] Abate, M., Patrizio, G., Finsler Metrics- A Global Approach. LNM 1591, Springer-Verlag, Berlin Heidelberg, 1994. [2] Bács´ o, S. and Cheng, X., Finsler conformal transformations and the curvature invariances. Publ. Math. Debrecen, 70/1-2(2007). [3] Bao, D., Chern, S. S. and Shen, Z., An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, Inc., 2000. [4] Bismut, J-M. and Zhang, W., An Extension of a theorem by Cheeger and M¨ uller. Astérisque, 205, Soc. Math. Franch, Paris, 1992. [5] Bryant, R. L., Projectively flat Finsler 2-spheres of constant curvature. Selecta Math. (New Series), 3(1997), 161-203. [6] Chern, S. S., Chen, W. and K. Lam, Lectures on Differential Geometry. Series on University Mathematics, Vol. 1, World Scientific Publishing Co. Pte. Ltd., 2000. [7] Liu, K. and Zhang, W., Adiabatic limits and foliations. The Milgram Festschrift. Eds. A. Adem et. al., Contemp. Math., 279(2001), 195-208. [8] Mo, X., An Introduction to Finsler Geometry. Peking University series in Math., Vol. 1, World Scientific Publishing Co. Pte. Ltd., 2006. [9] Mo, X., A new characterization of Finsler metrics with constant flag curvatrure 1. Front. Math. China, 6(2)(2011), 309-323. [10] Szab´ o, Z., Positive definite Berwald spaces (structure theorems on Berwald spaces). Tensor, N. S., 35(1981), 25-39. [11] Zhang, W., Lectures on Chern-Weil Theory and Witten Deformations. Nankai Tracts in Mathematics, Vol. 4, World Scientific Publishing Co. Pte. Ltd., 2001. Huitao Feng: School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, People’s Republic of China E-mail address: [email protected] Ming Li: School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, People’s Republic of China E-mail address: [email protected]