Volume comparison and its applications in Riemann–Finsler geometry

advances in mathematics128,306 328(1997)article no.AI971630Volume Comparison and Its Applicationsin Riemann Finsler GeometryZhongmin Shen*Department of Mathematical Science,Indiana University Purdue University at Indianapolis,Indianapolis,Indiana46202-3216Received May9,1996;accepted December30,1996We introduce a new geometric quantity,the mean covariation for Finsler metrics,and establish a volume comparison theorem.As an application,we obtain someprecompactness and finiteness theorems for Finsler manifolds. 1997Academic Press1.INTRODUCTIONEarly work in conjunction with the global comparison geometry of Finsler manifolds was done by L.Auslander[A],who proved that if a complete Finsler n-manifold(M,F)satisfies Ric M (n&1),then the diameter diam M ?.In particular,?1(M)is finite.This is the Bonnet Myers theorem if F is Riemannian.Further,he proved that if a complete Finsler manifold(M,F)has non-positive flag curvature,then the exponential map exp x:T x MÄM is a local C1-diffeomorphism.This is the Cartan Hadamard theorem if F is Riemannian.We point out that Auslander used the Cartan connection and stated the above theorems under slightly stronger conditions on curvatures,because he did not realize that the cur-vature terms in the second variation formula can be simplified to the flag curvature.See[AZ2,BCh]for simplified formulas.We remark that the flag curvature is independent of a particular choice of connections,and hence is of particular interest in the metric geometry of Finsler manifolds. See[L,AZ1,D1 D3,GKR,Mo1,Mo2,K1,K2,BCh],etc.,for other interesting work on global Finsler geometry.In Riemann geometry,the Bishop Gromov volume comparison theorem [BCr,GLP]plays a very important role in the global differential geometry of Riemann manifolds.One of its applications is the Gromov precompact-ness theorem for the class C(n,*,d)with respect to the Gromov Hausdorff distance d GH,where C(n,*,d)denotes the set of all Riemann manifolds (M,g)satisfying Ric M (n&1)*and diam M d.For=>0,R 1,consider *E-mail:zshenÄ.3060001-8708Â97 25.00Copyright 1997by Academic PressAll rights of reproduction in any form reserved.a sub-class C =,R (n ,*,d )/C (n ,*,d ),which consists of (M ,g )satisfying that any =-ball B (x ,=)is contractible in B (x ,R=)/M .It can be shown that if two manifolds M 0,M 1#C =,R (n ,*,d )are sufficiently close with respect to d GH ,then they have to be homotopy bining this with the pre-compactness of C (n ,*,d ),one can conclude that there are only finitely many homotopy types in C =,R (n ,*,d )([P1])(see also [GPW]for a homeomorphism version of this).This is a generalized version of Cheeger's finiteness theorem [C1,C2]and Grove Petersen's finiteness theorem[GP].See also [Y,Z]for some other interesting finiteness theorems.Here we shall not mention the references on the convergence theory of Riemann metrics,because we will only focus on the finiteness problem.The reader is referred to a recent paper of P.Petersen [P3]for a beautiful discussion and references on the convergence of Riemannian metrics.Petersen [P1,P2]studied the finiteness problem for metric spaces in great generally.This enables us to find geometric (curvature)conditions and establish a finiteness theorem for a much larger class of manifolds equipped with ``smooth''inner metrics.First we shall briefly discuss inner metrics which naturally lead to Finsler metrics.Let d be an inner metric on a C n-manifold M .Suppose it satisfies the following Lipschitz condition:at each point x #M ,there is a coordinate neighborhood .:V /R n ÄU /M such thatA &1|x 0&x 1| d (.(x 0),.(x 1)) A |x 0&x 1|,\x 0,x 1#V ;(1.1a)|d (.(x 0),.(x 0+y )&d (.(x 1),.(x 1+y )|A |x 0&x 1||y |,\x 0,x 1#V ,y #B n (=).(1.1b)We can prove that (1.1)implies that the following limit exists:F d (x ,y ):=lim r Ä0d (.(x ),.(x +ry ))|r | 0,\x #V ,y #R n (1.2)Further,F (x ,y )=F d (x ,y )has the following properties:F (x ,y )=|r |F (x ,y ),\r #R ,y #R n ;(1.3)F (x ,y )=0if and only if y =0;(1.4)|F (x 0,y )&F (x 1,y )| A |x 0&x 1||y |,\x 0,x 1#V ,y #R n ;(1.5)F (x ,y 1+y 2) F (x ,y 1)+F (x ,y 2),\y 1,y 2#R n .(1.6)A function F :TM Ä[0, )is called a Lipschitz Finsler metric if it satisfies (1.3) (1.6)in local coordinates.One can show that if d =d F is the inner metric induced by a Lipschitz Finsler metric F ,then d satisfies (1.1)307RIEMANN FINSLER GEOMETRYand reproduces F=F d by(1.2)([BM];see also[V,DCP]for the study of inner metrics satisfying(1.1a)only).Thus there is an one-to-one corre-spondence between Lipschitz inner metrics and Lipschitz Finsler metrics. To study the differential geometry of Finsler manifolds,we shall restrict ourselves to C Finsler metrics.A function F:TMÄ[0, )is called a C Finsler metric if it satisfies(1.3)and the following convexityÂregularity conditions:(1.7)F is smooth on TM"[0];(1.8)the Hessian(g ij(x,y)),y{0,is positive definite,whereg ij(x,y):=122F2y y(x,y).Note that(1.7),(1.8)imply(1.4),(1.5)and(1.6).The inner metric d=d F induced by F reproduces F=F d by(1.2).By the homogeneity of F,one always has F(x,y)=-g ij(x,y)y i y j.F is called Riemannian if g ij(x)= g ij(x,y)are independent of y.F is called locally Minkowskian if g ij(x,y)= g ij(y)are independent of x.Let F x denote the restriction of F onto T x M.When F is Riemannian, (T x M,F x)are all isometric to the Euclidean spaces R n.But,for general Finsler metric F,(T x M,F x)may be not isometric to each other.Thus the geometry of Finsler manifolds becomes more complicated.The Finsler metric F induces a familly of inner products g v in T x M byg v(u,w)=g ij(x,y)u i w j,(1.9) where v=y i(  x i)|x,u=u i(  x i)|x,w=w i(  x i)|x.Take an arbitrary basis[e i]ni=1for T x M.Let B x(1)=[y=(y i):F(y i e i) 1].Put g vij =g v(e i,e j).Define the mean distortion+:TM"[0]Ä(0, )by+(v):=vol(B x(1))vol(B(1))-det(g vij).(1.10)Clearly,+(v)is independent of a particular choice of[e i]ni=1,and+(*v)=+(v),\*{0.The mean covariation H:TM"[0]ÄR is defined byH(v):=ddt[ln+(#v)]|t=0,(1.11)where#v is the geodesic with#v(0)=v.Put H(0)=0.It is easy to see thatH(*v)=*H(v),\*#R.308ZHONGMIN SHENRoughly speaking,the mean covariation H(v )measures the average rate of changes of (T x M ,F x )in the direction v #T x M .We say |H M | +if |H(v )| +F (v ),\v #TM "[0].An important property is that H=0for Finsler manifolds modeled on a single Minkowski space.In particular,H=0for Berwald spaces.Locally Minkowski spaces and Riemann spaces are all Berwald spaces.In order to introduce other geometric quantities,one needs (linear)con-nections.In Finsler geometry,there are several important linear connec-tions such as the Berwald connection [Be],the Cartan connection [C],and the Chern connection [Ch1,Ch2,BCh].See [S]for a new interesting linear connection and its relationship with others.All of them are uniquely determined by two equations (torsion equation and metric equation)and reduce to the common one when F is Riemannian.Except for the Cartan connection,all of them are torsion-free.But the Cartan connection is metric-compatible.In a Finsler manifold (M ,F ),one cannot get a connec-tion which is torsion-free and metric-compatible,unless F is ing any of these connections,one can define three curvatures R ,P ,Q (Q =0for torsion free connections).These curvature terms depend on a particular choice of linear connections.The flag curvature tensor (defined by R only)is independent of a particular choice of these connections,that is,the term appears in the second variation of length,thus is of particular interest to us.We remark that if F is Riemannian,then P =Q =0,and the flag curvature tensor is the Riemannian curvature tensor.The average of the flag curvature tensor is the Ricci curvature Ric:TM ÄR .It has the property that Ric(rv )=r 2Ric(v ),r >0,v #TM .We say Ric M (n &1)*if Ric(v ) (n &1)*F (v )2,\v #TM .More details will be given in Section 2.For a constant *#R and + 0,putV *,+(r ):=Vol(S n &1(1))|r0e +t s *(t )n &1dt ,(1.12)where s *(t )denotes the unique solution to y "+*y =0with y (0)=0,y $(0)=1.Note that V *,+(r )=vol(B n (r ))(1+o (r ))as r Ä0+.We shall show that vol(B (x ,r ))=vol(B n (r ))(1+o (r ))as r Ä0+.Thuslimr Ä0+vol(B (p ,r ))V *,+(r )=1.(1.13)The following is our main theorem.Theorem 1.1.Let (M ,F )be complete Finsler manifold .Suppose thatRic M (n &1)*,|H M | +.(1.14)309RIEMANN FINSLER GEOMETRYThen for any0<r<R,vol(B(x,R)) V*,+(R)vol(B(x,r))V*,+(r).(1.15)In particular,vol(B(x,r)) V*,+(r).(1.16) In Section6,we shall discuss the case when the equality in(1.15)holds. Given n,*,+ 1,let M(n,*,+)denote the class of pointed complete Finsler n-manifold(M,p,F)satisfying the bounds(1.14).As a direct conse-quence of Theorem1.1,we have the followingCorollary 1.2.The class M(n,*,+)is precompact in the pointed Gromov Hausdorff topology.This corollary follows from Theorem1.1and[GLP,P2].A function\:[0,r)Ä[0, )is called a contractibility function if it satisfies(i)\(0)=0,(ii)\(=) =,(iii)\(=)Ä0as=Ä0,(iv)\is non-decreasing.Given a contractibility function\,a metric space X is said to be LGC(\)if for every=#[0,r]and x#X,the ball B(x,=)is contractible inside B(x,\(=)).For a number r>0,if every ball B(x,=)is contractible inside B(x,=),x#X,0<=<r,namely,X is LGC(\)for\(=)==:[0,r)Ä[0,r),then we say X has contractibility radius c(X) r.Given n,*, +,\,d,let M(n,*,+,d,\)denote the class of compact LGC(\)Finsler n-manifolds satisfyingRic M (n&1)*,|H M| +,diam M d.(1.17)Corollary 1.3.The class M(n,*,+,d,\)contains only finitely many homotopy types.In[C1,C2],J.Cheeger proved that if a compact Riemannian manifold M n satisfies the bound4 K M *,diam M d,vol M v,then the injec-tivity radius inj M i o(n,*,4,d,v)>0.Thus M is LGC(\)space for \(=)==:[0,i o]Ä[0, ).In[GP],Grove and Petersen proved that if a compact Riemannian manifold M n satisfies the bounds K M *,diam M d, vol M v,then every ball B(x,=)is contractible in B(x,R=)for\= =o(n,*,d,v)and R=R(n,*,d,v).Thus M is a LGC(\)space for \(=)=R=:[0,=o]Ä[0, ).It is an interesting question under what cur-vature bounds is a compact Finsler manifold M n satisfying the bounds diam M d,vol M v a LGC(\)space for some\depending only on those bounds.This problem will be discussed somewhere else.We remark that 310ZHONGMIN SHENboth arguments in [C1]and [GP]were carried out using Toponogov's comparison theorem.However,we can show that,in a Finsler manifold,Topogonov's triangle comparison theorem does not hold,unless it is Riemannian.In other words,a Finsler manifold is not curved from below in the sense of Alexandrov [A1,A2]unless it is Riemannian.By the same argument as in [GLP],one can obtain the followingCorollary 1.4.Let (M ,F )be a complete Finsler n-manifold satisfying the bounds (1.17).Then the first Betti number b 1(M ) c (n ,*,+,d ).There are many other important theorems in Riemannian geometry are false for Finsler manifolds.For example,the Cheeger Gromoll splitting theorem [CG]is no longer true,even for flat Finsler manifolds (say Minkowski spaces).Nevertheless,Milnor's theorem still holds for certain Finsler manifolds.Corollary 1.5.Let (M ,F )be a complete Finsler n-manifold withRic M 0,H M =0.Then any finitely generated subgroup 1of the fundamental group ?1(M )has polynomial growth of order n .2.PRELIMINARIESIn this section we shall recall some basic facts of Riemann Finsler geometry.See [M,R,BCh]for more details.We begin with the simplest Finsler manifolds.Let V be a vector space,and let F o :V Ä[0, )be a function satisfying (i)F o (*y )=|*|F (y ),(ii)F o is C on V "[0],and (iii)g ij (y ):=12( F 2o  y i y j )(y )is positive definite for y {0.F o is called a Minkowski norm on V ,and (V ,F o )is called a Minkowski space .Each T x V is naurally identified with V .Thus F o induces a Finsler metric F on V .There are lots of Minkowski spaces,but there is a unique Euclidean space,up to an isometry.Let (M ,F )be an arbitrary Finsler manifold.By definition,the restriction F x of F to T x M is a Minkowski norm in T x M .(T x M ,F x )is called a Minkowski tangent space at x #M .In general,(T x M ,F x )are not linearly isometric to each other.It is possible that on (M ,F ),we have infinitely many distinct Minkowski tangent spaces.There is no notion of angles between two tangent vectors in a Finsler manifold.Nevertheless,the volume form is well-defined [B1].Let [e i ]n i =1be a local basis for TM and ['i ]n i =1be its dual basis for T *M .Put 311RIEMANN FINSLER GEOMETRYB x(1):=[y=(y i):F(y i e i) 1].B x(1)is a strictly convex open subset in R n.Let B n(1)denote the standard unit ball in R n.The volume form dv is defined bydv=vol(B n(1))vol(B x(1))'17}}}7'n,(2.1)where vol(A)denotes the Euclidean volume of a subset A/R n.dv is inde-pendent of a particular choice of positive basis[e i]ni=1!As usual,thevolume vol(U)of an open subset U/M is defined by vol(B(x,r))= U dv. It is easy to see that the r-ball B(x,r)in a Minkowski space has the same volume of B n(r)in the Euclidean space R n.Busemann[B1]proved that for any bounded open subset U/M,vol(U)=H d(U),where H d(U) denotes the Hausdorff measure of U with respect to d=d F.This fact might be true for Lipschitz innerÂFinsler metrics(compare[V]).In order to define curvatures,it is more convenient to consider the pull-back tangent bundle than the tangent bundle,because our geometric quantities depend on directions.Let TM o=TM"[0]and let?*TM denote the pull-back of the tangent bundle TM by?:TM oÄM.Denote vectors in?*TM by(v;w),v#TM o, w#T?(v)M.For the sake of simplicity,we denote by i|v=(v;  x i|x), v#T x M the natural local basis for?*TM.The Finsler metric F defines two tensors g and A in?*TM byg( i|v, j|v)=g ij(x,y),A( i|v, j|v, k|v)=12F(x,y)g ijy k(x,y),where v=y i(  x i)|x.g and A are called the fundamental and Cartan tensors,respectively.Note that(?*TM,g)is a Riemannian vector bundle.A trivial fact is that F is Riemannian if and only if A=0.In Finsler geometry,we study connections and curvatures in(?*TM,g), rather than in(TM,F).The pull-back tangent bundle?*TM is a very special vector bundle.It has a unit vector l defined byl v=1F(v)(v;v).It is easy to see that A(X1,X2,X3)=0whenever X i=l for some i=1,2,3.Let[E i]ni=1be a local frame for?*TM.Define the dual co-frame[|i]ni=1on TM o by(v;?*(X))=|i(X)E i,X#T(TM o). Put l=l i E i,A ijk=A(E i,E j,E k),g ij=g(E i,E j).312ZHONGMIN SHENWe have the followingTheorem 2.1(Chern).There is a unique set of local 1-forms [|j i ]1 i ,j non TM o such that d|i =|j 7|j i (2.2)dg ij =g kj |i k +g ik |j k +2A ijk |n +k ,(2.3)where |n +k :=dl k +l l |l k +l k d (ln F ).Further ,[|i ;|n +i ]is a local co-frame for T *(TM o ).In a standard local coordinate system (x i ;y i )in TM ,take a natural basis [E i = i ]n i =1for ?*TM .We have|j i =1i jk (x ,y )dx kwhere1i jk =#i jk +1Fg il [A jks N s l &A ljs N s k &A kls N s j ]N s l =#s la y a &1Fg sj A jlk #k ab y a y b (2.4)#i jk =12g il { x j g lk + x k g lj & x l g jk =.We call [|j i ]the set of local connection forms .It defines a linear connec-tion {in ?*TM by{X Y =[XY i +Y j |j i (X )]E i ,X #T (TM o ),Y =Y i E i #C (?*TM ).(2.5)We also get two bundle maps \,+:T (TM o )Ä?*TM ,defined by\:=|i E i ,+:=F|n +i E i (2.6)Note that the VTM :=ker \is the vertical tangent bundle of TM o .Put HTM :=ker +.We have the direct composition T (TM o )=HTM ÄVTM .Tangent vectors in HTM are called horizontal ,and tangent vectors in VTM are called vertical .An important fact is that \|HTM and +|VTM are bundle isomorphisms.Define the set of local curvature forms 0j i by0j i :=d|j i &|j k 7|k i .313RIEMANN FINSLER GEOMETRYBy(2.2),one can show that0j i does not have vertical part,that is,one can write0j i=1R j i kl|k7|l+P j i kl|k7|n+l.Define the curvature tensors R,P in?*TM byR(U,V)W=u k v l w j Rj iklE i,P(U,V)W=u k v l w j P j i kl E i,where U=u i E i,V=v i E i,W=w i E i#?*TM.A Finsler manifold(M,F)is called a Berwald space if P=0.Let_=span[u,v]/T x M be a two-dimensional section.The flag cur-vature K(_;v)of the flag[_,v]is defined byK(_;v):=g(R(U,V)V,U)g(V,V)g(U,U)&g(U,V)2,where U=(v;u),V:=(v;v)#?*TM.When F is a Riemannian,K(_)=K(_;v) is independent of v#_,that is,the sectional curvature in Riemannian geometry.Further,it is independent of the above-mentioned linear connec-tions.Thus it really does not matter which connection should be used in the metric geometry of Finsler manifolds.Fixing a unit vector v#T x M,let[e i]ni=1,e n=v,be a basis for T x Msuch that[(v;e i)]ni=1is an orthonormal basis for?*TM.Let_i=span[e i,v],1 i n&1.The Ricci curvature Ric(v)is defined byRic(v):=:n&1i=1K(_i;v)F(v)2.The linear connection{in(2.5)defines the covariant derivative D v u of a vector field u on M in the direction v#T x M as follows.Let c be a curve in M with c(0)=v.Let c=dcÂdt be the canonical lift of c in TM o.Let u(t)=u|c(t)and U(t):=(c;u(t))#?*TM.Define D v u by(v;D v u):={dcÂdt U(0).(2.7) Note that D satisfies all properties of linear connections in TM,except forthe linearity in v,that is,D v1+v2u{D v1u+D v2u.Thus D is not a linear con-nection in TM in a usual sense.A vector field u=u(t)along c is called parallel if D dcÂdt u=0.A curve#:[0,a]ÄM is a geodesic if and only if#is parallel along#,i.e.,D##=0.(2.8) In this case,#must be parametrized proportional to arc-length.314ZHONGMIN SHENThe exponential map exp x :T x M ÄM is defined as usual,that is,exp x (v )=#v (1),where #v is a geodesic with #v (0)=x and #(0)=v .The Hopf Rinow theorem says that if (M ,d F )is complete,then exp x is defined on all of T x M for all x #M .This implies that any two points x 0,x 1#M can be joined by a minimizing geodesic.From now on,we always assume that (M ,d F )is complete.It is a well-known fact that the exponential map exp x is C away from the origin in T x M and only C 1at the origin with exp x |0=identity [W].In [B2],Busemann proved that if exp x is C 2at the origin for all x #M ,then all (T x M ,F x )are isometric to each other.On the other hand,Ichijyo[I]proved that in a Berwald space,all (T x M ,F x )are isometric to each other.Finally,Akbar Zadeh [AZ3]proved that exp x is C all over T x M for all x #M ,if and only if the Finsler metric is Berwald.Using the exponential map exp x ,one can easily show that d 2x =d (x ,})2is C near x and C 1at x .We remind the reader that (F x )2is only C 1at the origin,although exp x is C in a Berwald space.Proposition 2.2.If d 2x is C 2at x ,then F is Riemannian at x .Proof .Let .:V /R n ÄU /M be a local coordinate system at x with .(0)=x .Let h (z )=d 2x b .(z ),z #V .Note that h (z ) h (0)=0,z #V .Byassumption h is C 2in V .Thush (z )=12 2h z i zj (0)z i z j +o (|z |).As we have pointed in Section 1,d =d F reproduces F =F d by (1.2),that is,F (x ,y )2=lim r Ä0h (ry )2r 2=12 2h z i z j (0)y i y j .By definition,F is Riemannian at x .KThe cut-value t v of a vector v #T x M is defined to be the largest number r >0such that #v is minimizing on [0,r ].Let I x :=[v #T x M ,F (v )=1].The map v #I x Ät v #[0, )is continuous.The cut-locus C x =[exp p t v v :v #I x ]has zero Hausdorff measure in M .The injectivity radius inj x at x is defined by inj x =inf v #I x t v .Let 0x :=M "C x ,and O x =[(t ,x ):t <t v ].Then exp x :O x Ä0x is a diffeomorphism (C 1at the origin).For v #T x M ,define R v :T x M ÄT x M byR v (u )=R (U ,V )V ,(2.9)where U =(v :u ),V =(v ;V )#?*TM .315RIEMANN FINSLER GEOMETRYFix v#T x M,and let#v be a normal geodesic from x with#v(0)=v.Along#v,we have a family of inner products g t=g#v(t)in T#v (t)M(see(1.9))and flag curvatures R t=R#v(t):T#v (t)MÄT#u(t)M.For the sake ofsimplicity,we shall denote D t=D#v ,if no confusion is caused.(2.3)impliesthat for any vector fields u=u(t),w=w(t)along#v,ddt(g t(u(t),w(t))=g t(D t u(t),w(t))+g t(u(t),D t v(t)).(2.10) A vector field J=J(t)along#v is called a Jacoby field if it satisfiesD t D t J(t)+R t J(t)=0.(2.11)Lemma 2.3.A vector field Jualong#v with J u(0)=0and D t J u(0)=u is a Jacobi field if and only ifJ u(t)=d(exp x)|tv(tu).(2.12) In particular,a vector field J given by(2.12)is smooth along#v.We shall always denote by#v the geodesic with#v(0)=v#T x M and by J u the Jacobi field along#v defined by(2.12).Lemma 2.4(Gauss Lemma).For every Jacobi field Ju,J u(t)is per-pendicular to#v(t)for all t with respect to g t,if and only if u is perpendicular to v with respect to g v.From(2.12),we can see that exp x is singular at rv#T x M if and only if there is0{u#T x M,such that the Jacobi field J u satisfies J u(r)=0.In this case we call#v(r)a conjugate point with respect to x.By the standard argu-ment,one can show that if the flag curvature K(_;v) *,then there are no conjugate points on#|[0,r),for r=?Â-*(= if* 0).This is the so-called Cartan Hadamard theorem in Finsler geometry[A].The index lemma is still true.For a vector field W=W(t)along#v with W(0)=0,defineI(W,W)=|r0[g t(D t W(t),D t W(t))&g t(R t W(t),W(t))]dt.Lemma 2.5(Index Lemma).Suppose that#v does not contain conjugatepoints on[0,r].Let J be a Jacobi field along#v with J(0)=0.Then for any vector field W along#with W(0)=0and W(r)=J(r),I(J,J) I(W,W),the equality holds if and only if W=J.For a Jacobi field J u,we haveI(J u,J u)=g r(D t J u(r),J u(r)).By a standard argument and the index lemma,one can easily prove Bonnet and Myers'theorem for Finsler manifolds([A]).A Finsler space(M,F)is said to be modeled on a single Minkowski spaceif for every geodesic#,the parallel translation P t0,t1:(T#(t)M,F#(t))Ä(T#(t1)M,F#(t1))is an isometry for all t0,t1.In this case,all(T x M,F x)arelinearly isometric to each other.The class of such manifolds contains all Riemann manifolds and locally minkowski manifolds.Define F*:T R nÄ[0, )byF*\y i x i}x+= :n i=1(y i)2+* :i=1(y i)4,(2.13)where*:R nÄ[0, )is a C function.Clearly,(R n,F*)in(2.13)is not modeled on a single Minkowski space,if*:R nÄ[0, )is not constant. See[AIM]for more interesting Finsler metrics from physics.Proposition 2.6.If(M,F)is modeled on a single Minkowski space, then H=0.Proof.By definition,if u=u(t)is a parallel vector field along a geodesic#v,then F(u(t))=constant.Let[e i(t)]ni=1be a parallel basis for T#v(t)Mwith e i(0)=e i,1 i n.We haveB#v (t)(1):=[(y i):F(y i e i(t)) 1]=B x(1).We also have det(g t(e i(t),e j(t)))=det(g v(e i,e j)).Thus+(#v(t))=constant. This implies H(v)=0.KProposition 2.7([I]).Any Berwald space is modeled on a single Minkowski space.3.THE SINGULAR RIEMANN METRICS g x AND g xThe Finsler metric F induces a singular Riemann metric g x on T x M"[0] byg x(u,w):=g v(u,w),\u,w#T tv(T x M)t T x M,v#T x M.(3.1)Let I x=[v#T x M:F(v)=1].Let g x be the induced Riemann metric of g x on I x/T x M.Regard T x M as the cone C(I x)over I x.By the homogeneity of F,we haveg x=dt2Ät2g x.(3.2) Let'x denote the volume form of g x on T x M"[0]=C(I x)"[o].Let[e i]ni=1be a basis for T0(T x M)t T x M.Extend it to a global basis forT(T x M).Let[%i]ni=1be the dual basis for T*(T x M).We have'x|tv=-det(g x(e i,e j))|tv%17}}}7%n=-det(g(e i,e j))%17}}}7%n.(3.3) Let'x denote the volume form of g x on I x.By(3.2)we have'x|tv=t n&1'x|v7dt,\(t,v)#C(I x).(3.4) Define the density3x at x#M by3x=vol(I x,'x)vol(S n&1(1)).(3.5)When F is Riemannian,3x=1,\x#M.In general,3x does not have to be constant unless F is a weak Landsberg space(see[BS]).It is still an open problem whether or not there are two universal constants0<a n<b n, n=dim M such thata n 3xb n,\x#M.(3.6) If the norm of the Cartan tensor A is small enough,say&A& -3Â10,then (3.6)holds for some a n,b n.Define a smooth Riemannian metric g x inside0x"[x]byg x|#v (t)=g#v(t)(3.7)(see(1.9)).Putg x:=(exp x)*g x.(3.8) In general,g x is singular at the origin,unless F is Riemannian.Regard exp x as a map C(I x)ÄM.By the Gauss Lemma,we haveg x=dt2Äh t,(3.9) where h t is a family of Riemannian metrics on I x.Lemma 3.1.The metric g x satisfies1 t2h tÄg x,12th ttÄg x.(3.10)Proof.Let u,w#T v(I x M)/T tv(T x M)t T x M.Let g t=g#(t).Clearly,g t(u(t),w(t))Äg x(u,w),(3.11) where u(t),w(t)are parallel vector fields along#with u(0)=u,w(0)=w. For u#T x M,let J u(t)=d(exp x)|tv(tu).We haveh t(u,w)=g x(d(exp x)|tv(tu),d(exp x)|tv(tw))=g t(J u(t),J w(t)).(3.12) Thus by(2.10)htt(u,w)=g t(D t J u(t),J w(t))+g t(J u(t),D t J w(t)).(3.13)Put J u=(1Ât)J u(t),for t{0.Since J is smooth along#and J(0)=0, then J u:=(1Ât)J u(t)converges,as tÄ0.Put J(0)=lim tÄ0J u(t).Then J u is smooth.Thus D t J u is bounded.Observe that D t J u(t)=J u(t)+tD t J u(t).We can conclude that1tJ u(t)ÄD t J u(0)=u.(3.14) Now(3.10)follows from(3.11) (3.13).KThe following proposition is important in computing the flag curvature.Proposition 3.2.(i)Let D x be the Levi Civita connection of g x in0x. The along any normal geodesic#v,v#I x,D xt=D t.(3.15)(ii)Let u#T v(I x),and let_t=span[d(exp x)|tv(u),#v(t)]/T#v(t)M.Let K x(_t)denote the sectional curvature of g x.Then the flag curvature K(_t,#v(t))of F satisfiesK(_t;#v(t))=K x(_t)=&122t2(h t)uu+14t(h t)u:(h t):;t(h t)+;(h t)uu,(3.16)where[e:]n&1:=1is a basis for T v(I x),(h t)uu=h t(u,u),(h t)u:=h t(u,e:),and(h t):;=h t(e:,e;).Proof .Let (t ,x )be the conical (or polar)coordinate system in0x =exp x (O x ),regarding T x M as C (I x ).Let (t ,x;s ,y )be the standard coordinate system in T0x .By the Gauss lemma or (3.9),we have\100(g x ):;(t ,x )+=\100g :;(t ,x ;1,0)+=\100(h t ):;(t ,x )+.Put D x  t (  x :)=#;t:(t ,x)(  x ;).We have #;t:(t ,x )=12(h t );&(t ,x ) t(h t ):&;(t ,x ).We can put D (  t )(  x :)=1;t:(t ,x;1,0)(  x ;).By (2.4),1;t:(t ,x ;1,0)=#;t:(t ,x )&(h t );&(t ,x )A :&{(t ,x ;1,0)#{tt (t ,x).An easy computation yields that #{tt (t ,x)=0.Thus 1;t:(t ,x ;1,0)=#;t:(t ,x).This implies (3.15).Let R x denote the Riemann curvature tensor of g x on 0x .Note that J :(t ):=d exp x |(t ,x )(t (  x :)=t (  x )|(t ,x )is a Jacobi field of both F and g x .By the Jacobi equation (2.11),one obtains(R x )t x :=&1t D x  t D x  t J :(t )=&1t D  t D  t J :(t )=R t x :.By an easy computation,one obtain that for _t =span [  t ,  x :],K x (_t )=& t [g t (D  t J :(t ),J :(t ))]&g t (D  t J :(t ),D  t J :(t ))(g t )(J :(t ),J :(t ))=&12 2 t (h t )::+14 t (h t ):;(h t );& t (h t )&:(h t )::.K See [LS]for further discussions on Riemann metrics with conical singularities.。