24(2008), 103–114 www.emis.de/journals ISSN 1786-0091
DUAL CONNECTIONS IN FINSLER GEOMETRY
TETSUYA NAGANO AND TADASHI AIKOU
Abstract. In the present paper, we generalize the notion ofstatistical structureand its dual connection in Riemannian geometry to Finsler ge- ometry. We shall show that the Berwald connectionDof a Finsler manifold is a statistical structure. In particular, as an application of this fact, we shall show that, if thehh-curvature of the Berwald connectionDvanishes identically, then the given Finsler metric induces a Hessian metric on the base manifold.
1. Introduction
LetM be a connected smooth manifold of dimM =nwith a (semi-) Riemann- ian metrich. Astatistical structure in (M, h) is a symmetric linear connection D satisfying the Codazzi equation (DXh)(Y, Z) = (DYh)(X, Z) for all vector fields X, Y and Z onM, that is, Dh is totally symmetric. As is well-known, for an arbitrary symmetric (0,3)-tensor fieldC, the (1,2)-tensor fieldS defined by h(S(X, Y), Z) = C(X, Y, Z) induces a statistical structure D on (M, h) by D=∇ −S/2.
A linear connectionD∗ on (M, h) defined by
Xh(Y, Z) =h(DXY, Z) +h(Y, D∗XZ)
is called thedual connectionofD. IfDis a statistical structure, then its dualD∗ is also a statistical structure on (M, h), and we have the relation∇= (D+D∗)/2 for the Levi-Civita connection ∇ of (M, h) (cf. [2]). Recently the geometry of statistical structure becomes an interesting topics in differential geometry. In particular, the interest in statistical structure arises from the study of affine geometry and Hessian geometry (e.g., [7] and [10]). In [2], it is proved that the
2000Mathematics Subject Classification. 53B40.
Key words and phrases. Finsler connections, Dual connections.
The second author is supported in part by Grant-in-Aid for Scientific Research No.
17540086(2007), The Ministry of Education, Science Sports and Culture.
103
flatness of a statistical structureDimplies the existence of a Hessian metric on the base manifoldM.
Since Finsler geometry includes Riemannian geometry as a special case, it is natural to generalize the notions of statistical structure and the dual connection in Finsler geometry. The aim of the present paper is to define a statistical structure D and itsdual connection D∗ in a Finsler manifold.
In Finsler geometry, the Chern connection ∇ is a standard tool for study- ing Finsler manifolds, since∇ satisfies the metrical condition and the symmet- ric property (see Definition 3.1 below). In particular, if the given metric is a Riemannian metric, then ∇ is just the Levi-Civita connection of the given Riemannian metric. In the second and third section, we shall review some fun- damental facts in Finsler geometry from [1].
On the other hand, the Berwald connectionD satisfies the symmetric prop- erty, but not the metrical condition. The Berwald connectionD, however, plays an important role for some topics in Finsler geometry. In particular, we shall show that the Berwald connection is a statistical structure in this generalized sense (Theorem 5.1). In the last section, as an application of this fact, we shall show that, if the hh-curvature of the Berwald connection vanishes identically, then the base manifoldM admits a flat statistical structure in the original sense, and thusM is a Hessian manifold (Theorem 6.1).
2. Finsler metrics and Cartan tensor
Letπ:T M →M be the tangent bundle of a connected smooth manifoldM. We denote byv= (x, y) the points inT M ify∈π−1(x) =TxM. We introduce a coordinate system on T M as follows. LetU ⊂M be an open set with local coordinate (x1, . . . , xn). By settingv=P
yi¡
∂/∂xi¢
xfor everyv∈π−1(U), we introduce a local coordinate (x, y) = (x1, . . . , xn, y1, . . . , yn) onπ−1(U).
Definition 2.1. A functionL:T M −→Ris called aFinsler metric onM if 1. L(x, y)≥0, andL(x, y) = 0 if and only if y= 0,
2. L(x, λy) =λL(x, y) for∀λ∈R+={λ∈R: λ >0}, 3. L(x, y) is smooth onT M× =T M\{0}
are satisfied. The pair (M, L) is called a Finsler manifold. For eachX ∈TxM, its normkXkis defined by kXk=L(x, X).
The differential π∗ of the submersion π:T M× → M induces an exact se- quence
(2.1) 0−→V −→i T(T M×)−→π∗ T Mg −→0,
whereV is thevertical subbundlewhich is locally spanned by{∂/∂yj}j=1,...,non π−1(U), andT Mg ={(y, v)∈T M××T M v ∈Tπ(y)M}is the pullback bundle ofT M byπ∗.
Since the natural local frame field {∂/∂xi}i=1,...,n on U is identified with the one of T Mg on π−1(U), we use the notation X = P
(∂/∂xi)⊗Xi for a sectionX ofT Mg. Furthermore, since kerπ∗=V, the morphismπ∗ is given by π∗=P
(∂/∂xi)⊗dxi.
On the other hand, since T Mg is naturally identified with V ∼= kerπ∗, any sectionX ofT Mg is considered as a section ofV. We denote byXV the section ofV corresponding toX ∈Γ(T Mg):
X =X ∂
∂xi ⊗Xi ⇐⇒ XV =X ∂
∂yi ⊗Xi.
A Finsler metricLis said to beconvex ifF =L2/2 isstrictly convex on each tangent spaceTxM, that is, the Hessian (Gij) defined by
(2.2) Gij(x, y) = ∂2F
∂yi∂yj
is positive-definite. In the sequel, we assume the convexity of L. Then T Mg admits a metricGdefined byG(X, Y) =P
GijXiYj for allX=P
(∂/∂xi)⊗Xi andY =P
(∂/∂xj)⊗Yj. We also set Cijk =1
2
∂Gij
∂yk = 1 4
∂3L2
∂yi ∂yj ∂yk. Then we define a symmetric tensor fieldC: ⊗3T Mg →Rby
(2.3) C(X, Y, Z) =X
CijkXiYjZk
for all sectionsX, Y, Z of T M. It is trivialg C vanishes identically if and only if Gis a Riemannian metric onM. This tensor fieldCis called theCartan tensor field.
The multiplier groupR+∼={cI∈GL(T M×); c∈R+} ⊂GL(T M×) acts on the total space by multiplication mλ: T M× 3 v = (x, y) → λv = (x, λy) ∈ T M× for∀λ∈R+. This action induces a canonical section E of V defined by E(v) = (v, v) for all v ∈T M. We shall consider E as a section of T Mg, and we denote it by the same notationE. This sectionEis called thetautological section ofT Mg. Then it is easily shown thatL=p
G(E,E) and
(2.4) C(E,·,·)≡0.
3. Chern connection
The vertical subbundleV of the submersionπ:T M× →M is uniquely deter- mined. A subbundle H ⊂T(T M×) complementary to V is called ahorizontal subbundle:
(3.1) T(T M×) =V ⊕T Mg ∼=V ⊕H.
An Ehresmann connection of π is a selection of horizontal subbundles. An Ehresmann connection is given by aT Mg-valued 1-formθsatisfying
(3.2) θ(XV) =X
for every sectionXofT M. If an Ehresmann connectiong θis given, the subbundle H := kerθis a horizontal subbundle. In the sequel of the present paper, we shall denote byAk(T Mg) the space of smooth T M-valuedg k-forms onT M×.
Since we are concerned with the tangent bundle, the bundle T Mg is also naturally identified with the horizontal subbundle H, and any section X ∈ A0(T Mg) is considered as a section of H. We denote by XH the section of H corresponding toX ∈A0(T Mg):
X =X ∂
∂xi ⊗Xi ⇐⇒ XH =X δ δxi ⊗Xi,
where we set{δ/δx1, . . . , δ/δxn}= (∂/∂xi)H. By the definitions above, we have (3.3) π∗(XH) =X, π∗(XV) = 0
and
(3.4) θ(XH) = 0, θ(XV) =X
for everyX ∈A0(T Mg).
According to the decomposition (3.1), we get the splitting d =dV ⊕dH of the differential operatordonT M×:
dHf(X) =XH(f) and dVf(Y) =YV(f)
for everyf ∈C∞(T M×). Also any covariant exterior derivation∇:Ak(T Mg)→ Ak+1(T M) ong T Mg has the splitting∇=∇H⊕ ∇V:
∇HXY =∇XHY and ∇VXY =∇XVY for allX, Y ∈A0(T Mg) respectively.
3.1. Definition of Chern connection. In this subsection, we shall recall the definition of Chern connection from [4] and [1].
Definition 3.1. TheChern connectionon (M, L) is a covariant exterior deriva- tion∇:Ak(T Mg)→Ak+1(T Mg) determined by the following conditions.
(1) ∇issymmetric:
(3.5) ∇π∗= 0.
(2) ∇isalmost G-compatible:
(3.6) ∇HG= 0,
where we take the Ehresmann connectionθ defined by
(3.7) θ=∇E.
Remark 3.1. From the assumption (3.5), we can easily show that theT Mg-valued 1-form θ defined by (3.7) is an Ehresmann connection. In [4] or [8], this θ is called the Cartan’s non-linear connection. It is known thatθ defined by (3.7) is uniquely obtained from the given Finsler metric L. The definition (3.7) ofθ and the homogeneity ofLgives
(3.8) ∇HXE= 0
for every X ∈ A0(T Mg). This equation means that the horizontal subbundle H = kerθis invariant by the action m•of R+. ¤
The assumption (3.5) is equivalent to
(3.9) ∇HXY − ∇HYX−π∗[XH, YH] = 0 and
(3.10) ∇VYX−π∗[YV, XH] = 0, and the assumption (3.6) is equivalent to
(3.11) dHG(X, Y) =G(∇HX, Y) +G(X,∇HY) for allX, Y ∈A0(T Mg).
3.2. TorsionT∇ and curvature R∇ of ∇. In this subsection, we shall recall the definitions of torsion and curvature of the Chern connection ∇ defined in the previous subsection. We also recall some propositions concerned with torsion and curvature (cf. [1]).
Definition 3.2. Thetorsion T∇∈A2(T Mg) of∇ is defined by
(3.12) T∇=∇θ.
Because of (3.3), we obtain T∇(XV, YV) = 0 for all X, Y ∈A0(T Mg). If we defineT∇HH(X, Y) :=T∇(XH, YH) andT∇HV(X, Y) :=T∇(XH, YV), then (3.3) and (3.4) give
(3.13) T∇HH(X, Y) =−θ[XH, YH] and
(3.14) T∇HV(X, Y) =∇HXY −θ[XH, YV] for allX, Y ∈A0(T Mg). The following is easily obtained.
Proposition 3.1. The horizontal part T∇HH and the mixed part T∇HV satisfy (3.15) T∇HH(X, Y) +T∇HH(Y, X)≡0
and
(3.16) T∇HV(X, Y)−T∇HV(Y, X)≡0
for allX, Y ∈A0(T Mg). Furthermore the mixed partT∇HV satisfies the following
(3.17) T∇HV(X,E) = 0.
Definition 3.3. Thecurvature R∇∈A2(End(T Mg)) of∇is defined by
(3.18) R∇=∇2.
Similarly to the torsionT∇, we obtainR(XV, YV)≡0 for allX, Y ∈A0(T Mg).
We set RHH∇ (X, Y)Z :=R(XH, YH)Z andRHV∇ (X, Y)Z :=R(XH, YV)Z. We list up some identities concerning withR∇.
The symmetry assumption (3.5) gives
Proposition 3.2. The horizontal part RHH∇ and the mixed part RHV∇ satisfy the followings:
(3.19) RHH∇ (X, Y)Z+RHH∇ (Y, Z)X+RHH∇ (Z, X)Y ≡0, (3.20) RHV∇ (X, Y)Z−RHV∇ (Z, Y)X≡0.
The definition ofT∇ impliesT∇=R∇E. Thus we get
Proposition 3.3. The curvature R∇ and the torsion T∇ satisfies the relation T∇=R∇E:
(3.21) RHH∇ (X, Y)E=T∇HH(X, Y), (3.22) RHV∇ (X, Y)E=T∇HV(X, Y).
The almostG-compatibility assumption (3.6) gives
Proposition 3.4. The curvature R∇ and the torsionT∇ satisfy the followings:
(3.23) G(RHH∇ (X, Y)Z, W)+G(RHH∇ (X, Y)W, Z)+2C(T∇HH(X, Y), Z, W) = 0 G(RHV∇ (X, Y)Z, W) + G(Z, R∇HV(X, Y)W) +
+ 2(∇HXC)(Y, Z, W) + 2C(T∇HV(X, Y), Z, W) = 0 (3.24)
This last identity gives
(3.25) RHV∇ (X,E) = 0
4. Dual connections
In this section, we shall introduce the notion of dual connection in Finsler geometry. Let (M, L) be a Finsler manifold, and θ the Ehresmann connection defined by (3.7). LetD=DH⊕dV be a symmetric Finsler connection satisfying
(4.1) DE=θ.
Under this assumption, the horizontal partTDHH of the torsion TD =Dθ coin- cides withT∇HH, and therefore we denote it byTHH in the sequel. Furthermore,
similarly to the case of Riemannian geometry[10], we callDastatistical structure of (T M , G) ifg
(4.2) (DXHG)(Y, Z) = (DHYG)(X, Z) is satisfied for allX, Y, Z∈A0(T Mg).
Definition 4.1. A Finsler connectionD∗ =D∗H ⊕dV is called the dual con- nection of a statistical structureD ifD∗ satisfies
(4.3) XHG(Y, Z) =G(DHXY, Z) +G(Y, D∗HX Z) for allX, Y, Z∈A0(T Mg).
Proposition 4.1. The dual connectionD∗ of a statistical structureD is sym- metric.
Proof. It is trivial that (DXHG)(Y, Z) is symmetric inY andZ, and thus we have (DHXG)(Y, Z) =XHG(Y, Z)−G(DHXY, Z)−G(Y, DHXZ)
=XHG(Z, Y)−G(DXHZ, Y)−G(Z, DXHY)
= (G(DHXZ, Y) +G(Z, D∗HX Y))−G(DXHZ, Y)−G(Z, DXHY)
=G(Z, D∗HX Y)−G(Z, DHXY)
and (DYHG)(X, Z) =G(Z, D∗HY X)−G(Z, DYHX). Consequently we have (DHXG)(Y, Z)−(DHYG)(X, Z)
=G(Z, D∗HX Y)−G(Z, DHXY)−G(Z, D∗HY X) +G(Z, DYHX)
=G(Z, D∗HX Y −DY∗HX)−G(Z, DHXY −DYHX)
=G(Z, D∗HX Y −DY∗HX)−G(Z, π∗[XH, YH])
=G(Z, D∗HX Y −DY∗HX−π∗[XH, YH])
=G(Z,(D∗π∗)(XH, YH))
for allX, Y, Z∈A0(T Mg). SinceGis nondegenerate, we have shown thatD∗ is symmetric if and only ifD is a statistical structure. ¤ From the definition (4.3), the following proposition is obtained immediately.
Proposition 4.2. Let D∗ be the dual connection of a statistical structure D.
Then we have
(4.4) DHG+D∗HG= 0,
and therefore the dualD∗ is also a statistical structure of(T M , G).g
Form Proposition 4.1 and 4.2, for a statistical structure D, the covariant exterior derivation (D+D∗)/2 satisfies the symmetric condition (3.5) and the almost G-compatibility (3.6), and thus the uniqueness of Chern connection ∇ gives∇= (D+D∗)/2. Therefore, we have
Theorem 4.1. Let D∗ be the dual connection of a statistical structure D of (T M , G). Then the Chern connectiong ∇ of(T M , G)g is given by
(4.5) ∇=1
2(D+D∗)
Furthermore the dual ofD∗ coincides with D, that is,D∗∗ =D.
LetRD∗ =D∗2 be the curvature of the dual connectionD∗. We investigate the relation between the curvaturesRHHD andRHHD∗ .
Theorem 4.2. Let D∗ be the dual connection of a statistical structure D of (T M , G).g
(1) The horizontal part RHHD andRHHD∗ of D andD∗ are related by (4.6) G(RHHD (X, Y)Z, W)+G(Z, RHHD∗ (X, Y)W)+2C(THH(X, Y), Z, W) = 0.
(2) The dual connection D∗ satisfies RDHH∗ = 0 if and only if D satisfies RHHD = 0.
Proof. From (4.2), we have [XH, YH]G(Z, W)
= [XH, YH]HG(Z, W) + [XH, YH]VG(Z, W)
=G(D[XH,YH]HZ, W) +G(Z, D∗[XH,YH]HW) + [XH, YH]VG(Z, W) for allX, Y, Z, W ∈Γ(T Mg). From (3.13) we note that
[XH, YH]VG(Z, W)
= (D[XH,YH]VG)(Z, W) +G(D[XH,YH]VZ, W) +G(Z, D[XH,YH]VW)
=−2C(THH(X, Y), Z, W) +G(D[XH,YH]VZ, W) +G(Z, D[X∗ H,YH]VW), sinceDV =D∗V =dV. Hence we obtain
[XH, YH]G(Z, W) =−2C(THH(X, Y), Z, W)
+G(D[XH,YH]Z, W) +G(Z, D∗[XH,YH]W).
On the other hand
XHYHG(Z, W) =G(DHXDHYZ, W) +G(DHYZ, D∗HX W) +G(DHXZ, D∗HY W) +G(Z, DX∗HDY∗HW)
implies
[XH, YH]G(Z, W) =G(DXHDYHZ−DHYDHXZ, W) +G(Z, DX∗HD∗HY W−DY∗HD∗HX W), and therefore we obtain (4.6).
We supposeRHHD = 0 (resp. RHHD∗ = 0). Then (4.1) implies THH(X, Y) =RHHD (X, Y)E=RD∗HH(X, Y)E = 0
for all X, Y ∈ Γ(T Mg). Hence we obtain RHHD∗ = 0 (resp. RHHD = 0) from
(4.6). ¤
5. Berwald connection
There exists another canonical Finsler connection which plays an important role for some topics in Finsler geometry. For the Ehresmann connection θ by (3.7), the Finsler connectionD=dV ⊕DH onT Mg defined by
(5.1) DHXY =θ[XH, YV]
is called theBerwald connection in a Finsler manifold (M, L). From the equa- tion (3.14) and the definition (5.1), we obtain a relation between the Chern connection∇and the Berwald connectionD:
(5.2) DHXY =∇HXY −T∇HV(X, Y)
Then, from (4.7) and (4.17), it follows that D defined by (5.1) satisfies (4.1), and thusD is a Finsler connection in the sense of previous section. Furthermore Ddefined by (5.1) symmetric, namely,Dπ∗= 0. In fact, from (3.14) and (3.16) we have
(Dπ∗)(XH, YH) =DXHY −DHYX−π∗[XH, YH]
=∇HXY −THV(X, Y)− ∇HYX+T∇HV(Y, X)−π∗[XH, YH]
= 0
and (Dπ∗)(XV, YH) =DVXY −π∗[XV, YH] = 0, sinceDV =dV.
The almostG-compatibility and the relation (4.2) induce the following (5.3) (DHXG)(Y, Z) =G(T∇HV(X, Y), Z) +G(Y, T∇HV(X, Z)).
On the other hand, if we setX =E in (3.24), then (3.20) and (3.22) give G(T∇HV(Z, Y), W) +G(Z, T∇HV(W, Y)) + 2(∇HEC)(Y, Z, W) = 0, and therefore (5.3) is equivalent to
(5.4) (DHXG)(Y, Z) =−2(∇HE C)(X, Y, Z), and from (4.4)
(5.5) (DX∗HG)(Y, Z) = 2(∇HE C)(X, Y, Z).
for allX, Y, Z∈A0(T Mg). Thus Proposition 4.1 implies
Theorem 5.1. The Berwald connection D and its dual connectionD∗ are sta- tistical structures on the Finsler bundle(T M , G).g
A Finsler manifold (M, L) is called a Landsberg space if the Berwald con- nection D coincides with the Chern connection ∇. In this case, from (4.5), we have
Proposition 5.1. If (M, L)is a Landsberg space, then the Berwald connection D and its dual D∗ coincide with the Chern connection∇.
6. Finsler manifolds satisfying RDHH = 0
The class of Berwald space which is characterized byRHV∇ = 0 orRHVD = 0 has been studied by [11], and the classification of Berwald spaces is obtained. If a Finsler manifold (M, L) satisfiesRHH∇ ≡0, then the metricGonT Mg induces a flat Riemannian metric onM, and thusM is locally Euclidean (cf.[1]).
On the other hand, the class of Finsler manifolds satisfying RHHD ≡ 0 has not been studied enough yet. In the sequel, we shall show that, if a Finsler manifold (M, L) satisfiesRHHD = 0, then the metricL induces a Hessian metric g on M. Here a Riemannian manifold (M, g) is said to be aHessian manifold if the following conditions are satisfied (cf. [10]).
(1) There exists a flat affine connectionD onM, (2) the metricg=P
gijdxi⊗dxj is given by the Hessian of some function ψ with respect to the affine coordinate (x1, . . . , xn) ofD, that is, g is given by the covariant derivativeDdψ of the 1-fromdψ:
gij= ∂2ψ
∂xi∂xj.
We suppose thatRHHD = 0. Then, the Ricci identityTD=RDEgivesTHH = 0, and thus the horizontal subbundleH is integrable. Hence there exists a section v: M →T M× satisfyingv∗θ= 0. Such a sectionvis called ahorizontal section (cf. [1]). Sincev satisfiesv∗◦dV = 0, we obtain
(6.1) d◦v∗=v∗◦dH.
Lemma 6.1. Suppose that the horizontal partRHHD of the Berwald connection Dvanishes identically. Then the induced connectionv∗D by a horizontal section v is a flat affine connection onM.
Proof. We denote byωjithe connection form ofDwith respect to the local frame field{∂/∂x1, . . . , ∂/∂xn}. If we set
RHHD ∂
∂xj =X ∂
∂xi ⊗Ωji,
the curvature formΩji is given byΩji =dHωij+P
ωil∧ωjl, and thus the assump- tion means thatΩji = 0.
On the other hand, the connection form ofv∗Dis given byv∗ωji. Hence, from (6.1), the curvature of v∗D is given by
d(v∗ωji) +X
(v∗ωji)∧(v∗ωjl) =v∗(dHωji+X
ωli∧ωlj) =v∗Ωji.
Consequentlyv∗D is flat. ¤
By Theorem 4.2 and Lemma 6.1, ifD satisfies RHHD = 0, then the induced connection v∗D∗ is also a flat affine connection on M. We set D = v∗D and D∗=v∗D∗. We also denote byg =P
Gij(x, v(x))dvi⊗dvj =P
gijdxi⊗dxj the induced Riemannian metric v∗G on M. Then the condition (4.3) implies the following relation:
(6.2) Xg(Y, Z) =g(DXY, Z) +g(Y, D∗XZ)
for all X, Y, Z ∈ Γ(T M). Furthermore, D and D∗ are symmetric affine con- nections such that Dg and D∗g are totally symmetric. Therefore, if RHHD = 0 is satisfied, then (M, g, D) and (M, g, D∗) are flat statistical manifolds. As is well-known (cf. [2]), if (M, g, D) is flat, thenM is locally Hessian.
Theorem 6.1. If the curvature RHHD of the Berwald connection D vanishes identically, thenM admits a flat statistical structure, and thereforeM is locally a Hessian structure(D, g).
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Tetsuya Nagano,
Department of Informatics, Siebold University of Nagasaki, Nagasaki, 851-2195 Japan E-mail address:[email protected] Tadashi Aikou,
Department of Mathematics and Computer Science, Faculty of Science, Kagoshima University,
Kagoshima, 890-0065 Japan
E-mail address:[email protected]
