Some Open Problems in Finsler Geometry

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1 Some Open Problems in Finsler Geometry provided by Zhongmin Shen March 8, 2009 We give a partial list of open problems concerning positive definite Finsler metrics. 1 Notations and Definitions A Finsler metric on a manifold M is a function F : TM [0, ) which has the following properties (a) F is C on TM {0}; (b) F (x, λy) =λf (x, y), λ>0, (d) For any tangent vector y T x M, the following bilinear symmetric form g y : T x M T x M R is positive definite, g y (u, v) := 1 2 [ ] F 2 (x, y + su + tv) s=t=0. 2 s t F is said to be reversible, iff (x, y) =F (x, y). Let We have g ij (x, y) = 1 2 [F 2 ] yi yj (x, y). g y (u, v) =g ij (x, y)u i v j. Let F be a Finsler metric on a manifold M. For an oriented curve C M from p to q, b ( ) L F (C) := F c(t), ċ(t) dt, a 1

2 where c :[a, b] C M is a parameterization of C with c(a) =p and c(b) =q. For two points p, q M, define d(p, q) := inf C L F (C), where the infimum is taken over all curves C from p to q. Let d be the induced distance function of a Finsler metric F. Then for any y T x M and any C 1 map c :[0,ɛ) M with c(0) = x, ċ(0) = y, d(x, c(t)) F (x, y) = lim. (1) t 0 + t Conversely, given an arbitrary distance function d on a manifold, if it is nice enough, then it determines a Finsler metric by (1). We have the following diagram F d F. f-product Manifolds: Let (M i,α i ), i =1, 2, be arbitrary Riemannian manifolds and M = M 1 M 2. Let f :[0, ) [0, ) [0, ) be an arbitrary C function satisfying f(λs, λt) =λf(s, t), (λ >0) and f(s, t) 0if(s, t) 0. Define ) F := f ([α 1 (x 1,y 1 )] 2, [α 2 (x 2,y 2 )] 2, (2) where x =(x 1,x 2 ) M and y = y 1 y 2 T (x1,x 2)(M 1 M 2 ) = T x1 M 1 T x2 M 2. Clearly, F has the following properties: (a) F (x, y) 0 with equality holds if and only if y =0; (b) F (x, λy) =λf (x, y), λ>0; (c) F (x, y) isc on TM \{0}. Now we are going to find additional condition on f = f(s, t) under which the matrix g ij := 1 2 [F 2 ] yi y j is positive definite. Take standard local coordinate systems (x a,y a )intm 1 and (x α,y α )intm 2. Then (x i,y j ):=(x a,x α,y a,y α ) is a standard local coordinate system in TM. Express α 1 (x 1,y 1 )= ḡ ab (x 1 )y a y b, α 2 (x 2,y 2 )= ḡ αβ (x 2 )y α y β, 2

3 where y 1 = y a x and y a 2 = y α x. We obtain α ) ( ) 2fss ȳ (g ij = a ȳ b + f s ḡ ab 2f st ȳ a ȳ β, 2f st ȳ b ȳ α 2f tt ȳ α ȳ β + f t ḡ αβ where ȳ a ) := ḡ ab y b and ȳ α := ḡ αβ y β. By an elementary argument, one can show that (g ij is positive definite if and only if f satisfies the following conditions: f s > 0, f t > 0, f s +2sf ss > 0, f t +2tf tt > 0, and In this case, f s f t 2ff st > 0. ) ( det (g ij = h [α 1 ] 2, [α 2 ] 2) ) ) det (ḡ ab det (ḡ αβ, where h := (f s ) n1 1 (f t ) n2 1{ f s f t 2ff st }, where n 1 := dim M 1 and n 2 := dim M 2. Locally minimizing curves can be parameterized by a regular map c(t) whose local coordinates (c i (t)) are determined by a system of 2nd order ordinary differential equations c i (t)+2g i (c(t), ċ(t)) = 0, where G i (x, y) are locally given by G i (x, y) := 1 { 4 gil (x, y) 2 g jl x (x, y) g } jk k x (x, y) y j y k. l Any such map c(t) is called a geodesic. It is easy to verify that any geodesic c(t) has a constant speed, i.e., F (ċ(t)) = constant. F is said to be positively complete if any geodesic is defined on a half-line (a, ). F is said to be complete, if any geodesic is defined on the whole line (, ). A Finsler metric F is said to be affinely equivalent to another Finsler metric F, if they have the same spray, i.e., their geodesic coefficients are related by G i (x, y) =G i (x, y). A Finsler metric F is said to be projectively equivalent to another Finsler metric F if they have the same geodesics as point sets, i.e., their geodesic coefficients are related by G i (x, y) =G i (x, y)+p (x, y)y i. 3

4 A Finsler metric F is said to be locally projectively flat if at every point, there is a local coordinate system (x i ) in which geodesics are straight lines as point sets, or equivalently G i (x, y) =P (x, y)y i or equivalently F xk y l(x, y)yk = F x l(x, y). Beltrami Theorem: A Riemannian metric is of constant sectional curvature if and only if it is locally projectively flat. A Finsler metric F on M is said to be (globally) projectively flat if there is a Riemannian metric α of constant sectional curvature on M such that F is pointwise projectively equivalent to α. A Finsler metric F is said to be locally dually flat if at every point, there is a local coordinate system (x i ) in which the function L := F 2 satisfies L x k y l(x, y)yk =2L x l(x, y). (3) A Finsler metric F is said to be (globally) dually flat if there is a positively y-homogeneous function of degree three H = H(x, y) ontm such that at every point, there is a local coordinate system (x i ) in which the function L := F 2 satisfies L x k y l(x, y)yk =2L x l(x, y) and H(x, y) = 1 2 L x k(x, y)yk. It is easy to verify that if a Riemannian metric L = g ij (x)y i y j satisfies (3), then there is a function ψ = ψ(x) such that g ij (x) =ψ x i x j (x). For each y T x M, define D y : C (TM) T x M by } D y V (x) := {dv i (y)+v j (x) Gi y (x, y) j x x. i D y V is called the covariant derivative of V in the direction y. D={D y } y TM\{0} has all the properties of an affine connection on M except for the linearity in the direction y. A vector field V (t) along a geodesic c(t) is said to be parallel if DċV (t) =0. A Finsler metric F is called a Berwald metric if D is an affine connection, i.e., D u+v W =D u W +D v W. 4

5 1.1 Cartan torsion For each y T x M {0}, define C y : T x M T x M T x M R by C y (u, v, w) := 1 3 [ ] F 2 (x, y + su + tv + rw) s=t=r=0. 4 s t r C := {C y } y TM\{0} is called the Cartan torsion. Fact: C = 0 if and only if F is Riemannian. Define I y (u) := where g ij (x, y) =g y (e i,e j ). n g ij (x, y)c y (e i,e j,u), ij=1 Fact (Deicke): I = 0 if and only if F is Riemannian. The bound of the Cartan torsion C at a point x M is defined by C x := sup sup F x(y)=1 v T xm C y (v,v,v). [g y (v, v)] 3 2 Set C := sup x M C x. A Finsler metric is said of bounded Cartan torsion if C <. 1.2 Distortion Let F be a Finsler metric on a manifold M. Let {e i } n i=1 be a basis for T xm and {ω i } n i=1 the dual basis for T x M. Denote by dµ x := σ(x)ω 1 ω n the Busemann volume form at x, where σ(x) := Vol(B n ) Vol{(y i ),F(y i e i ) < 1}. For each y T x M {0}, define [ det(g y (e i,e j )) ] τ(x, y) := ln. σ(x) The scalar function τ : TM \{0} R is called the distortion. Fact: τ = 0 if and only if F is Riemannian. 5

6 1.3 Landsberg Curvature Let F be a Finsler metric on a manifold M. Fory T x M {0}, let c(t) be the geodesic with ċ(0) = y. Define L y (u, v, w) := d [ ] Cċ(t) (U(t),V(t),W(t)) t=0, (4) dt where U(t),V(t),W(t) are parallel along c with U(0) = u, V (0) = v, W(0) = w. L := {L y } y TM\{0} is called the Landsberg curvature. Define n J y (u) := g ij (x, y)l y (e i,e j,u), i=1 where g ij (x, y) :=g y (e i,e j ). J := {J y } y TM\{0} is also an important quantity. F is called a Landsberg metric if L =0. Fact: If F is Berwaldian, then it is Landsbergian. A local formula for L y (u, v, w) =L ijk (x, y)u i v j w k : 3 G l L ijk = FF y l y i y j y l. 1.4 S-Curvature and E-Curvature Let F be a Finsler metric on a manifold M. Define S(x, y) := d [ ( )] τ c(t), ċ(t) t=0. (5) dt The scalar function S : TM \{0} R is called the S-curvature. S has the following property: S(x, λy) =λs(x, y), λ > 0. F is said to have isotropic S-curvature if there is a scalar function c(x) on M such that S(x, y) =(n +1)c(x) F (x, y), y TM \{0}. F is said to have constant S-curvature if c(x) =constant. In this case, we write S = constant. Define E y (u, v) := 1 2 [ ] S(x, y + su + tv) 2 s t. (6) s=t=0 E := {E y } y TM\{0} is called the E-curvature. E has the following property: E λy = λ 1 E y, λ > 0. 6

7 Fact: For Berwald metrics, S = 0 and E =0. In a local coordinate system, E y (u, v) =E ij (x, y)u i v j is given by E ij = 1 2 ( G m ). 2 y i y j y m Let E := g ij E ij. E is an important non-riemannian quantity. For each two-plane P T x M and y P, define E(P, y) :=F 3 E y (v, v) (x, y) F 2 (x, y)g y (v, v) [g y (y, v)] 2, (7) where P = span{y, v}. F is of isotropic E-curvature if there is a scalar function c(x, y) such that E(P, y) =c(x, y), y P T x M. It is equivalent to saying that { } E y (u, v) =(n +1)c(x, y) F 3 (x, y) F 2 (x, y)g y (u, v) g y (y, u)g y (y, v). Clearly, if F is of isotropic S-curvature, then it is of isotropic E-curvature. There is another important non-riemannian quantity. For y T x M {0}, let c(t) be the geodesic with ċ(0) = y. Define H y (u, v) := d [ ] Eċ(t) (U(t),V(t)) t=0, (8) dt where U(t),V(t) are parallel along c with U(0) = u, V (0) = v. In a local coordinate system, H y (u, v) =H ij (x, y)u i v j is given by H ij = E ij m y m. Let H := g ij H ij. H is also an important non-riemannian quantity. 7

8 1.5 Riemann Curvature and Ricci Curvature Let G = y i x i 2Gi (x, y) y i be a spray (induced by a Finsler metric). The Riemann curvature R y = R i k dxk x i p : T p M T p M is defined by R i k =2 Gi x k 2 G i yj x j y k +2Gj 2 G i y j y k Gi G j y j y k. (9) The Riemann curvature has the following properties: for any non-zero vector y T p M, R y (y) =0, g y (R y (u), v) =g y (u, R y (v)), u, v T p M. Let c(t) be the geodesic with ċ(0) = y. Take a family of geodesics c s (t) = H(s, t) with c 0 (t) =c(t) and let J(t) := H s (0,t). We have the following Jacobi equation DċDċJ(t)+Rċ(t) (J(t)) = 0. For each two-plane P T x M and y P, define K(P, y) := g y (R y (v),v) F 2 (x, y)g y (v, v) g y (y, v)g y (y, v). where P = span{y, v)}. F is said to be of scalar curvature or isotropic if there is a scalar function κ(x, y) such that K(P,y) =κ(x, y) is independent of two-planes P containing y. It is equivalent to saying that { } R y (u) =κ(x, y) F 2 (x, y)u g y (y, u) y. Fact: Every locally projectively flat Finsler metric is isotropic. But the converse is not true. The Ricci curvature Ric : TM R is defined by Ric(x, y) := trace of R y. F is said to be Ricci-isotropic if there is a scalar function κ = κ(x) such that Ric =(n 1)κ(x) F 2. 8

9 F is said to be Ricci-constant if there is a constant κ such that Ric =(n 1)κ F 2. We have the following volume comparison theorem: For any positively complete Finsler manifold (M,F) satisfying the bounds Ric (n 1)κF 2, S (n 1)δF, the volume of metric balls B(p, r) satisfies Vol(B(p, r)) Vol(S n 1 (1)) r 0 [ s κ (t)e δt] n 1 dt, where s κ (t) is the solution of y (t)+κy(t) =0,y(0) = 0,y (0) = Funk Metrics Let (V,ϕ) be a Minkowski space and B := {v V, ϕ(v) < 1}. y T x B=V, define F (x, y) > 0by x + y F (x, y) B. For each F is called the Funk metric on B. The Funk metric has the following properties: (a) S(x, y) = n+1 2 F (x, y); (b) E y (u, v) = n+1 4 F 3 (x, y) (c) L y (u, v, w) = 1 2 F (x, y) C y(u, v, w). { } F 2 (x, y)g y (u, v) g y (y, u)g y (y, v) ; (d) The Busemann-Hausdorff measure µ(b,f)=vol(b n (1)). More precisely, for any ball B(x, r) :={p B, d(x, p) <r} of radius r, r µ(b(x, r)) = Vol(S n 1 (1)) [e n+1 2(n 1) t sinh(t/2) ] n 1dt. 1/2 (e) R y (u) = { } F 2 (x, y)u g y (y, u) y, hence Ric(x, y) = 1 4 (n 1)F 2. (f) All geodesics are straight lines and every geodesic c(t) with ċ(0) ] = y T x B is defined on ( h, ), where h = 1 F (x,y) [1+F ln (x, y)/f ( y). Remark: (a)-(d) are proved by the author recently. The proof of (e) is given by Okada. By (a)(d)(e), we know that the volume comparison is sharp. 9

10 1.7 Eigenvalues For a compact Finsler manifold (M,F), the first eigenvalue is defined by M λ 1 (M,F) := inf [F (df )] 2 dµ f C (M) inf c R M f c 2 dµ, where dµ denotes the Busemann-Hausdorff volume form of F. Let Co (M) denote the space of all f C (M) with M fdµ = 0 and Ho 2(M) the completion of C o in the Hilbert space Ho 2(M). For f H2 (M), define E(f) := M [F (df )] 2 dµ M f. 2 dµ The critical values of E are discrete and divergent to, λ 1 <λ 2 < <λ i. They are called the eigenvalues of (M,F). The set C i of functions f Ho 2 (M) corresponding to λ i is a finite-dimensional cone. m i := dim C i is called the multiplicity of λ i. 10

11 2 Open Problems Problem 1 Is there any Finsler metric F with the following properties (a) F is non-berwaldian; (b) F is Landsbergian (L = 0) or weakly Landsbergian (J = 0); Note that for any regular (α, β)-metric F = αφ(β/α), L = 0 if and only if it is a Berwald metric. There are singular Landsberg (α, β)-metrics which are not Berwaldian. G. Asanov gave the first example. Z. Shen has classified all Landsberg (α, β)-metrics. Z.I. Szabo has claimed that every Landsberg metric must be Berwaldian. But there is a gap in his proof. Problem 2 Is there any (positively complete) Finsler metric with the following properties (a) F has zero flag curvature (K = 0); (b) F is Landsbergian (L = 0) or weakly Landsbergian (J = 0); (c) F is non-berwaldian. If there is a Finsler metric satisfying (a)-(c), then Problem 1 is solved. In dimension two, there are singular Finsler metrics with K = 0 and J =0, depending on a function of two variables (claimed by Bryant). Problem 3 There are non-berwaldian Randers metrics with S = 0. Is there any reversible non-berwaldian S-metrics? The classification of (α, β)-metrics with S = 0 has been completed by Cheng- Shen. S. Deng has found a non-berwaldian metrics with S =0. Problem 4 Is there any positively complete Finsler metric with the following properties (a) F has zero flag curvature (K = 0); (b) F has bounded Cartan torsion (sup C < ); (c) F is not locally Minkowskian. If a Finsler metric satisfying (a)-(b) is complete, then it is locally Minkowskian by a theorem of Akbar-Zadeh. Study and characterize all positively complete metrics on R n with K =0 (and sup C < ). Study and characterize complete Finsler metrics on R n with K =0. 11

12 Problem 5 Study and characterize complete Finsler metrics F on R n with K 1, S =(n +1)cF, where c = constant. If an additional condition sup I < is imposed, then F must be Riemannian. Without completeness, the Funk metric satisfies the above curvature conditions. We do not know any complete non-riemannian reversible Finsler metric on R n with K 1 and S =0. Problem 6 Study complete Finsler manifolds with the following properties (a) K 0; (b) J =0; (c) S =0; (d) I is bounded. Let (N,h) be any complete Riemannian surface with nonpositive sectional curvature. Let M = S 1 N or R N with the following Finsler metric F (x, y) = u 2 + h( x, ȳ) 2 + ɛ u 4 + h( x, ȳ) 4, (ɛ 0) where x =(t, x) M and y = u t +ȳ T xm =R T x N. F is a complete Berwald manifold with K 0 and bounded Cartan torsion. Since F is Berwaldian, we conclude that S =0,L = 0 and hence J =0. Problem 7 Classify Randers metrics of scalar flag curvature. Find Randers metrics with the following properties: a) it is not locally projectively flat, b) it is not of weakly isotropic flag curvature. B. Chen and L. Zhao have found an example. Problem 8 A Finsler metric is called an (α, β)-metric if it is in the form F = αφ(β/α) where α is a Riemannian metric and β is a 1-form. Find sufficient and necessary conditions on φ, α and β such that an (α, β)-metric F has constant flag curvature. Try to find some (α, β)-metrics of constant flag curvature. Bao-Robles-Shen have classifed Randers metrics F = α + β of constant curvature, among them are non-projectively flat ones. Shen-Yildirim have classed all projectively flat F =(α + β) 2 /α of constant curvature. Linfeng Zhou has proved that every F =(α+β) 2 /α of constant flag curvature must be projectively flat. Then one obtains a classification theorem on such metrics. We conjecture that there are no other types of (α, β)-metrics of constant flag curvature. Problem 9 Classify two-dimensional projectively flat (α, β)-metrics. 12

13 Problem 10 Classify (singular) (α, β)-metrics such that the induced Riemannian metric on its indicatrix is of constant sectional curvature. Problem 11 Study and characterize (α, β)-metrics with isotropic E-curvature. Determine those not of almost isotropic S-curvature. Problem 12 Study Finsler metrics in the following form F = {a i1 i 2p (x)y i1 y i2p } 1 2p. (10) Finsler metrics in the above form are called pth root-metrics. We are particularly interested in the fourth-root metrics F = 4 a ijkl (x)y i y j y k y l. The class of fourth-root metrics contains Riemannian metrics. Let α = a ij (x)y i y j be a Riemannian metric. Then F = 4 α 4 is a fourth-root metric, where α 4 := a ijkl (x)y i y j y k y l, a ijkl = 1 4! (pqst)=(ijkl) a pq (x)a st (x). (a) Find non-riemannian metrics of scalar flag curvature in the above form (10), (b) Find non-berwaldian Landsberg metrics in the above form (10), (c) Show that a fourth-root metric is of scalar flag curvature in dimension n 3 if and only if it is Riemannian. Problem 13 Study Finsler metrics in the following form F = a ijkl (x)y i y j y k y l + b ij (x)y i y j. (11) There are lots of projectively flat metrics in the above form. There are lots of Berwald metrcs in the above form. For example, the following family of metrics are Berwaldian. F = f(x, y)2 (u 2 + v 2 ) 2 + w 4 + ɛf(x, y)(u 2 + v 2 )+ɛw 2, where ɛ = constant and f(x, y) isac function. (a) Find non-riemannian metrics of constant flag curvature in the form (11), (b) Find Landsberg metrics in the form (11). 13

14 Problem 14 Study Finsler metrics in the following form F = a ijkl (x)y i y j y k y l + b ij (x)y i y j + c i (x)y i. (12) If c i 0, then F is irreversible. There are lots of projectively flat metrics of constant flag curvature in the form (12). (a) Classify metrics of constant flag curvature in the form (12). (b) Find Landsberg metrics in the form (12). Problem 15 Is there any Finsler metric with the following properties: (a) it is not locally projectively flat; (b) it is not a Randers metric; (c) it is of scalar flag curvature. Problem 16 Find (locally or globally) dually flat Finsler metrics of constant flag curvature. Try to classify these metrics. Note that the Funk metric is the only known example. Cheng-Shen-Zhou have classified locally dually flat Randers metrics of constant flag curvature. Problem 17 Is there any manifold which admits a (positively) complete Finsler metric of scalar curvature, but does not admit any (positively) complete Finsler metric of constant flag curvature. What are the topological obstructions for a compact manifold to admit a Finsler metric of scalar curvature? Problem 18 Study two-dimensional Finsler metrics with W o = 0. First verify that if K = 0, then W o = 0. How about the case when K = constant? Problem 19 Is there any Douglas metric which is not locally projectively Berwaldian (i.e., locally projective to a Berwald metric)? Is there any Douglas metric which is locally projectively Berwaldian, but not globally projectively Berwaldian? Problem 20 Show that a Finsler metric is of scalar flag curvature if and only if it is locally projectively equivalent to a Finsler metric of zero flag curvature. Problem 21 Study Finsler metrics which are of zero flag curvature. It is known that any Finsler metric of zero flag curvature on a compact manifold is locally Minkowskian. 14

15 Problem 22 Is there any Ricci-constant Finsler metric in dimension three which are not of constant flag curvature? Is there any Ricci-constant (α, β)- metric in dimension three which are not of constant flag curvature? Is there any Ricci-constant Finsler metric on S 1 S 2 whose flag curvature is not a constant? (asked by D. Bao) Problem 23 If a Finsler metric F is Ricci-isotropic, Ric =(n 1)c(x)F 2,is the scalar function c(x) a constant in dimension n 3? The answer is YES for Randers metrics. This is proved by C. Robles in her thesis, It is natural to study (α, β)-metrics with Ric =(n +1)c(x)F 2. Problem 24 Characterize Ricci-flat Berwald metrics. Note: In dimension n = 2, 3, Ricci-flat Berwald metrics must be locally Minkowskian. Proof: Every Berwald metric F = F (x, y) is affinely equivalent to a Riemannian metric α = α(x, y). If the Berwald metric F is Ricci-flat, then α is Ricci-flat. In dimension n =2, 3, α is flat. Thus F is R-flat. Since F is a Berwald metric, it is locally Minkowskian. Problem 25 Let (V,F) be a Minkowski space and F the induced Finsler metric on the indicatrix S = F 1 (1). Is the flag curvature of F positive everywhere? If F is of constant flag curvature K = 1, is F Euclidean? Problem 26 Let (V,F) be a Minkowski space and F the induced Finsler metric on the indicatrix S = F 1 (1). Suppose that the geodesics of F are the intersection curves of S with hyperplanes passing through the origin. Is F Euclidean? Let (V,F ) be dual to (V,F). Let S := F 1 (1) and F denote the induced Finsler metric on S. Is Vol(S, F ) equal to Vol(S, F ), where Vol denotes the Busemann volume of the Finsler metric? Problem 27 Is there any reversible or non-reversible Finsler metric F on S 2 such that the first eigenvalue λ 1 (S 2,F) > 8 Area(S 2,F)? It is known that for any Riemannian metric g on S 2, λ 1 (S 2 8,g) Area(S 2,g). Problem 28 Study the (non-linear) spectra of Finsler spaces of constant flag curvature. Are there any pair of isospectral, but not (locally) isometric, Finsler spaces of constant flag curvature? Problem 29 Does every compact Finsler manifold of negative Ricci curvature have discrete isometry group? For a compact Finsler manifold, under what curvature conditions, the isometry group is discrete? 15

16 Problem 30 Find all conformal invariants of a Finsler metric. Study the local metric structure of locally conformally flat Finsler metrics (with constant flag curvature). Here we say two Finsler metrics F 1 and F 2 on M are said to be conformally equivalent to each other if F 1 (x, y) =e φ(x) F 2 (x, y), y T x M. Problem 31 (Yamabe Problem). For a Finsler metric on a manifold, the scalar curvature r(x) at a point x is defined to be the average of the Ricci curvature on the unit tangent sphere at x. Given an arbitrary Finsler metric F = F (x, y) on a closed manifold, is there a scalar function ρ = ρ(x) such that F = e ρ(x) F (x, y) has constant scalar curvature. Problem 32 If a Finsler metric F = F (x, y) on an open domain Ω R n is of constant flag curvature, is there a non-trivial function ρ = ρ(x) such that F = e ρ(x) F (x, y) is of constant flag curvature? Note: if F = F (x, y) is a projectively flat Riemannian metric, then there are non-trivial functions ρ = ρ(x) such that F = e ρ(x) F (x, y) is of constant sectional curvature. Problem 33 Is there a Finsler metric on S 2 S 2 with positive flag curvature K > 0? It is a famous problem in Riemannian geometry to prove the (non- )existence of Riemannian metrics on S 2 S 2 with positive sectional curvature. Problem 34 In a Finsler manifold (M,F), define a parallel translation P c : T p M T q M along a curve c(t),a t b, by P c (V (a)) := V (b), where V (t) =V i (t) x i c(t) is a parallel vector field along c defined by { dv i ( )} Gi DċV (t) := (t)+ ċ(t),v(t) dt y j x i c(t) =0. The holonomy group at a point p is defined to be the group of parallel translations along loops at p. Classify the holonomy groups of Finsler manifolds defined by this parallel translations. Is there any holonomy group of a Finsler manifold which is not a holonomy group of any Riemannian manifold? Is there a Finsler manifold whose holonomy group is not a Lie group? Problem 35 Is there a notion of curvature B in Finsler geometry such that for a Finsler manifold (M,F), the curvature B 0 if and only if the space has non-positive curvature in the sense of Busemann, i.e., at any point p M, there is a neighborhood U p of p in which any pair of unit speed geodesics γ(t),σ(t), 0 t ɛ, with γ(0) = σ(0) = p satisfy ( ) d γ(s),σ(t) 1 ( ) 2 d γ(2s),σ(2t), 0 s, t ɛ. 16

17 Problem 36 Find a sufficient condition on curvatures for a complete reversible Finsler space to be Gromov hyperbolic. Problem 37 Study Finsler manifolds whose flag curvature K(P, y) = K(P ) is independent of the direction y P for all P T x M. Such metrics are called Finsler metrics of sectional flag curvature. Finsler metrics of constant flag curvature are of sectional flag curvature. B Chen and L Zhao have made some progress on this problem. Problem 38 Let Ω be an open subset in R n with a regular boundary Ω. Let F be a Finsler metric on Ω and d = d(x, Ω) the distance function to the boundary. Study the singular set of d. Problem 39 Study Finsler metrics satisfying H ij =(n +1)γF 1 h ij or H = (n 2 1)γF 1, where γ is a 1-form. H ij := E ij m y m, H := g ij H ij. Problem 40 Characterize Randers metrics with H ij =0orH =0. Problem 41 Study Finsler metrics of scalar flag curvature with K = 3θ F + σ, where θ is a 1-form and σ is a scalar function on the manifold. Problem 42 A Finsler metric F on a manifold M is called a W-quadratic metric if the coefficients of the Weyl curvature W i k = W j i kl (x)yj y l are quadratic in y at every point x M. Similarly we can define R-quadratic metrics using the Riemann curvature. It is easy to see that R-quadratic metrics are always W-quadratic metrics. Study local and global properties of R-quadratic or W- quadratic metrics. Characterize (α, β)-metrics which are R-quadratic or W- quadratic metrics. Problem 43 Study and characterize the solutions H = H(x, y) =P (x, y) + if (x, y) to the following equations on U R n : H x k = HH y k, H(x, λy) =λh(x, y), λ > 0 such that F (x o,y) is a Minkowski norm on T x ou = R n. Find some solutions expressed in terms of elementary functions. Let H = H(x, y) be a function on U R n satisfy the following algebraic equation zh 2 = y 2 2 y, x H + x 2 H 2, where z = a + ib is a non-zero complex number, denotes the standard Euclidean norm and, its inner product. Then for x close to the origin, any solution H = H(x, y) satisfies H x k = HH y k. 17

18 Problem 44 Study Finsler metrics with constant principal curvatures. The principal curvatures are the eigenvalues of the Riemann curvature R y : T x M T x M. Find some examples. Problem 45 Prove the 1/4-pinching sphere theorem for Randers spaces. Problem 46 Study submanifolds in a Randers space (especially, Randers Minkowski spaces). In a three-dimensional Randers-Minkowski space, what kind of surfaces have constant Gauss curvature? Problem 47 For an arbitrary scalar function K = K(x, y)on TM with K(x, λy) = K(x, y), λ >0, does there exist a Finsler metric whose flag curvature is K? 18