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 : T M → [0, ∞) which has the
following properties
(a) F is C ∞ on T M − {0};
(b) F (x, λy) = λF (x, y), λ > 0,
(d) For any tangent vector y ∈ Tx M , the following bilinear symmetric form
gy : Tx M × Tx M → R is positive definite,
gy (u, v) :=
i
1 ∂2 h 2
F (x, y + su + tv) |s=t=0.
2 ∂s∂t
F is said to be reversible, if F (x, −y) = F (x, y).
Let
gij (x, y) =
1 2
[F ]yiyj (x, y).
2
We have
gy (u, v) = gij (x, y)ui vj .
Let F be a Finsler metric on a manifold M . For an oriented curve C ⊂ M
from p to q,
Z b LF (C) :=
F c(t), ċ(t) dt,
a
1
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 LF (C),
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 ∈ Tx M and any C 1 map c : [0, ) → M with c(0) = x, ċ(0) = y,
F (x, y) = lim
t→0+
d(x, c(t))
.
t
(1)
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 (Mi , αi), i = 1, 2, be arbitrary Riemannian manifolds and M = M1 × M2. Let f : [0, ∞) × [0, ∞) → [0, ∞) be an arbitrary C ∞
function satisfying
f(λs, λt) = λf(s, t), (λ > 0) and
Define
f(s, t) 6= 0 if (s, t) 6= 0.
r F := f [α1(x1 , y1)]2, [α2(x2 , y2)]2 ,
(2)
where x = (x1, x2) ∈ M and y = y1 ⊕y2 ∈ T(x1 ,x2 ) (M1 ×M2 ) ∼
= Tx1 M1 ⊕Tx2 M2 .
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) is C ∞ on T M \ {0}.
Now we are going to find additional condition on f = f(s, t) under which
the matrix gij := 12 [F 2]yiyj is positive definite. Take standard local coordinate
systems (xa , ya ) in T M1 and (xα , yα ) in T M2 . Then (xi , yj ) := (xa , xα, ya , yα )
is a standard local coordinate system in T M . Express
q
q
α1(x1 , y1) = ḡab (x1)ya yb ,
α2 (x2, y2) = ḡαβ (x2 )yα yβ ,
2
∂
α ∂
where y1 = ya ∂x
a and y2 = y ∂xα . We obtain
2fssȳa ȳb + fs ḡab
2fst ȳa ȳβ
gij =
,
2fst ȳb ȳα
2ftt ȳα ȳβ + ft ḡαβ
where ȳa := ḡab yb and ȳα := ḡαβ yβ . By an elementary argument, one can show
that gij is positive definite if and only if f satisfies the following conditions:
fs > 0,
ft > 0,
fs + 2sfss > 0,
ft + 2tftt > 0,
and
fs ft − 2ffst > 0.
In this case,
det gij = h [α1]2, [α2]2 det ḡab det ḡαβ ,
where
n
o
h := (fs )n1 −1(ft )n2 −1 fs ft − 2ffst ,
where n1 := dim M1 and n2 := dim M2 .
Locally minimizing curves can be parameterized by a regular map c(t) whose
local coordinates (ci (t)) are determined by a system of 2nd order ordinary differential equations
c̈i(t) + 2Gi (c(t), ċ(t)) = 0,
where Gi(x, y) are locally given by
n ∂g
o
1
∂gjk
jl
Gi(x, y) := gil (x, y) 2 k (x, y) −
(x,
y)
yj yk .
4
∂x
∂xl
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 F̃ 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
G̃i (x, y) = Gi (x, y).
A Finsler metric F̃ 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
G̃i(x, y) = Gi(x, y) + P (x, y)yi .
3
A Finsler metric F is said to be locally projectively flat if at every point,
there is a local coordinate system (xi ) in which geodesics are straight lines as
point sets, or equivalently
Gi (x, y) = P (x, y)yi
or equivalently
Fxk yl (x, y)yk = Fxl (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 (xi ) in which the function L := F 2 satisfies
Lxk yl (x, y)yk = 2Lxl (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) on T M such that at every
point, there is a local coordinate system (xi ) in which the function L := F 2
satisfies
Lxk yl (x, y)yk = 2Lxl (x, y)
and
1
H(x, y) = − Lxk (x, y)yk .
2
It is easy to verify that if a Riemannian metric L = gij (x)yi yj satisfies (3), then
there is a function ψ = ψ(x) such that
gij (x) = ψxi xj (x).
For each y ∈ Tx M , define Dy : C ∞ (T M ) → Tx M by
n
o ∂
∂Gi
Dy V (x) := dV i (y) + V j (x) j (x, y)
|x .
∂y
∂xi
Dy V is called the covariant derivative of V in the direction y. D = {Dy }y∈T M \{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.,
Du+v W = DuW + Dv W.
4
1.1
Cartan torsion
For each y ∈ Tx M − {0}, define Cy : Tx M × Tx M × Tx M → R by
Cy (u, v, w) :=
i
1 ∂3 h 2
F (x, y + su + tv + rw) |s=t=r=0 .
4 ∂s∂t∂r
C := {Cy }y∈T M \{0} is called the Cartan torsion.
Fact: C = 0 if and only if F is Riemannian.
Define
n
X
Iy (u) :=
gij (x, y)Cy (ei , ej , u),
ij=1
where gij (x, y) = gy (ei , ej ).
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
kCkx :=
sup
sup
Fx (y)=1 v∈Tx M
|Cy (v, v, v)|
3 .
[gy (v, v)] 2
Set kCk := supx∈M kCkx . A Finsler metric is said of bounded Cartan torsion if
kCk < ∞.
1.2
Distortion
Let F be a Finsler metric on a manifold M . Let {ei }n
i=1 be a basis for Tx M and
∗
1
{ωi}n
the
dual
basis
for
T
M
.
Denote
by
dµ
:=
σ(x)ω
· · · ωn the Busemann
x
x
i=1
volume form at x, where
σ(x) :=
Vol(Bn )
.
Vol{(yi ), F (yi ei ) < 1}
For each y ∈ Tx M − {0}, define
τ (x, y) := ln
h pdet(g (e , e )) i
y i j
.
σ(x)
The scalar function τ : T M \ {0} → R is called the distortion.
Fact: τ = 0 if and only if F is Riemannian.
5
1.3
Landsberg Curvature
Let F be a Finsler metric on a manifold M . For y ∈ Tx M − {0}, let c(t) be the
geodesic with ċ(0) = y. Define
Ly (u, v, w) :=
i
dh
Cċ(t)(U (t), V (t), W (t)) |t=0,
dt
(4)
where U (t), V (t), W (t) are parallel along c with U (0) = u, V (0) = v, W (0) = w.
L := {Ly }y∈T M \{0} is called the Landsberg curvature.
Define
n
X
Jy (u) :=
gij (x, y)Ly (ei , ej , u),
i=1
where gij (x, y) := gy (ei , ej ). J := {Jy }y∈T M \{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 Ly (u, v, w) = Lijk (x, y)ui vj wk :
Lijk = −F Fyl
1.4
∂ 3 Gl
.
∂yi ∂yj ∂yl
S-Curvature and E-Curvature
Let F be a Finsler metric on a manifold M . Define
i
dh S(x, y) :=
τ c(t), ċ(t) |t=0.
dt
(5)
The scalar function S : T M \ {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 ∈ T M \ {0}.
F is said to have constant S-curvature if c(x) = constant. In this case, we write
S = constant.
Define
i
1 ∂2 h
Ey (u, v) :=
.
(6)
S(x, y + su + tv)
2 ∂s∂t
|s=t=0
E := {Ey }y∈T M \{0} is called the E-curvature. E has the following property:
Eλy = λ−1 Ey ,
6
λ > 0.
Fact: For Berwald metrics, S = 0 and E = 0.
In a local coordinate system, Ey (u, v) = Eij (x, y)ui vj is given by
Eij =
1 ∂ 2 ∂Gm .
2 ∂yi ∂yj ∂ym
Let
E := gij Eij .
E is an important non-Riemannian quantity.
For each two-plane P ⊂ Tx M and y ∈ P , define
E(P, y) := F 3(x, y)
Ey (v, v)
,
2
F (x, y)gy (v, v) − [gy (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 ⊂ Tx M.
It is equivalent to saying that
n
o
Ey (u, v) = (n + 1)c(x, y) F −3(x, y) F 2(x, y)gy (u, v) − gy (y, u)gy (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 ∈ Tx M − {0}, let c(t) be the geodesic with ċ(0) = y. Define
Hy (u, v) :=
i
dh
Eċ(t) (U (t), V (t)) |t=0,
dt
where U (t), V (t) are parallel along c with U (0) = u, V (0) = v.
In a local coordinate system, Hy (u, v) = Hij (x, y)ui vj is given by
Hij = Eij|mym .
Let
H := gij Hij .
H is also an important non-Riemannian quantity.
7
(8)
1.5
Riemann Curvature and Ricci Curvature
Let
G = yi
∂
∂
− 2Gi (x, y) i
∂xi
∂y
be a spray (induced by a Finsler metric).
∂
Rik dxk ⊗ ∂x
i |p : Tp M → Tp M is defined by
Ri k = 2
The Riemann curvature Ry =
∂ 2 Gi
∂ 2 Gi
∂Gi
∂Gi ∂Gj
− yj j k + 2Gj j k −
.
k
∂x
∂x ∂y
∂y ∂y
∂yj ∂yk
(9)
The Riemann curvature has the following properties: for any non-zero vector
y ∈ Tp M ,
Ry (y) = 0,
gy (Ry (u), v) = gy (u, Ry (v)),
u, v ∈ Tp M.
Let c(t) be the geodesic with ċ(0) = y. Take a family of geodesics cs(t) =
H(s, t) with c0 (t) = c(t) and let J(t) := ∂H
∂s (0, t). We have the following Jacobi
equation
DċDċ J(t) + Rċ(t)(J(t)) = 0.
For each two-plane P ⊂ Tx M and y ∈ P , define
K(P, y) :=
gy (Ry (v), v)
.
2
F (x, y)gy (v, v) − gy (y, v)gy (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
n
o
Ry (u) = κ(x, y) F 2(x, y)u − gy (y, u) y .
Fact: Every locally projectively flat Finsler metric is isotropic. But the converse
is not true.
The Ricci curvature Ric : T M → R is defined by
Ric(x, y) := trace of Ry .
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
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))
Z
r
h
in−1
sκ (t)eδt
dt,
0
where sκ (t) is the solution of y00 (t) + κy(t) = 0, y(0) = 0, y0 (0) = 1.
1.6
Funk Metrics
Let (V, ϕ) be a Minkowski space and B := {v ∈ V, ϕ(v) < 1}. For each
y ∈ Tx B = V , define F (x, y) > 0 by
x+
y
∈ ∂B.
F (x, y)
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) Ey (u, v) =
n+1 −3
(x, y)
4 F
n
o
F 2(x, y)gy (u, v) − gy (y, u)gy (y, v) ;
(c) Ly (u, v, w) = − 12 F (x, y) Cy (u, v, w).
n
(d) The Busemann-Hausdorff measure µ(B, F ) = Vol(B (1)). More precisely,
for any ball B(x, r) := {p ∈ B, d(x, p) < r} of radius r,
Z rh
in−1
n−1
− n+1 t sinh(t/2)
µ(B(x, r)) = Vol(S
e 2(n−1)
(1))
dt.
1/2
0
(e) Ry (u) = − 14
n
o
F 2 (x, y)u − gy (y, u) y , hence Ric(x, y) = − 14 (n − 1)F 2.
(f) All geodesics are straight lines and every geodesic
c(t) with ċ(0) i= y ∈ Tx B
h
is defined on (−h, ∞), where h =
1
F (x,y)
ln 1 + F (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
1.7
Eigenvalues
For a compact Finsler manifold (M, F ), the first eigenvalue is defined by
R
∗
(df)]2 dµ
M [F
R
λ1 (M, F ) :=
inf
,
f ∈C ∞ (M ) inf c∈R
|f − c|2dµ
M
where dµ denotes the Busemann-Hausdorff volume form of F R.
Let Co∞ (M ) denote the space of all f ∈ C ∞(M ) with M fdµ = 0 and
2
Ho (M ) the completion of Co∞ in the Hilbert space Ho2 (M ). For f ∈ H 2(M ),
define
R
[F ∗(df)]2 dµ
E(f) := M R
.
f 2 dµ
M
The critical values of E are discrete and divergent to ∞,
λ1 < λ2 < · · · < λi −→ ∞.
They are called the eigenvalues of (M, F ). The set Ci of functions f ∈ Ho2 (M )
corresponding to λi is a finite-dimensional cone. mi := dim Ci is called the
multiplicity of λi .
10
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 ChengShen.
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 kCk < ∞);
(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 Rn with K = 0
(and sup kCk < ∞). Study and characterize complete Finsler metrics on Rn
with K = 0.
11
Problem 5 Study and characterize complete Finsler metrics F on Rn with
K ≤ −1,
S = (n + 1)cF,
where c = constant.
If an additional condition sup kIk < ∞ 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 Rn 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
q
p
F (x, y) = u2 + h(x̄, ȳ)2 + u4 + h(x̄, ȳ)4,
( ≥ 0)
∂
where x = (t, x̄) ∈ M and y = u ∂t
+ ȳ ∈ Tx M = R ⊕ Tx̄ 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
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
1
n
o 2p
F = ai1 ···i2p (x)yi1 · · · yi2p
.
(10)
Finsler metrics in the above form are called pth root-metrics. We are particularly
interested in the fourth-root metrics
q
F = 4 aijkl(x)yi yj yk yl .
p
The class of fourth-root metrics contains Riemannian metrics. Let α = aij (x)yi yj
√
4
be a Riemannian metric. Then F = α4 is a fourth-root metric, where
α4 := aijkl(x)yi yj yk yl ,
aijkl =
1
4!
X
apq (x)ast(x).
(pqst)=(ijkl)
(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
rq
F =
aijkl(x)yi yj yk yl + bij (x)yi yj .
(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.
qp
F =
f(x, y)2 (u2 + v2 )2 + w4 + f(x, y)(u2 + v2 ) + w2,
where = constant and f(x, y) is a C ∞ function.
(a) Find non-Riemannian metrics of constant flag curvature in the form (11),
(b) Find Landsberg metrics in the form (11).
13
Problem 14 Study Finsler metrics in the following form
rq
F =
aijkl(x)yi yj yk yl + bij (x)yi yj + ci (x)yi .
(12)
If ci 6= 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 Wo = 0. First verify
that if K = 0, then Wo = 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
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, 2003. 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 F̄ the induced Finsler metric
on the indicatrix S = F −1(1). Is the flag curvature of F̄ positive everywhere?
If F̄ is of constant flag curvature K = 1, is F Euclidean?
Problem 26 Let (V, F ) be a Minkowski space and F̄ the induced Finsler metric
on the indicatrix S = F −1(1). Suppose that the geodesics of F̄ 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 F̄ ∗ denote the induced
Finsler metric on S ∗ . Is Vol(S ∗ , F̄ ∗) equal to Vol(S, F̄ ), where Vol denotes the
Busemann volume of the Finsler metric?
Problem 27 Is there any reversible or non-reversible Finsler metric F on S2
such that the first eigenvalue
8
?
Area(S 2 , F )
λ1 (S2 , F ) >
It is known that for any Riemannian metric g on S 2 ,
λ1 (S2 , g) ≤
8
.
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?
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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 F1 and F2 on M are said to be
conformally equivalent to each other if F1(x, y) = eφ(x) F2(x, y), y ∈ Tx 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 F̃ = eρ(x) F (x, y)
has constant scalar curvature.
Problem 32 If a Finsler metric F = F (x, y) on an open domain Ω ⊂ Rn is of
constant flag curvature, is there a non-trivial function ρ = ρ(x) such that F̃ =
eρ(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
F̃ = eρ(x) F (x, y) is of constant sectional curvature.
Problem 33 Is there a Finsler metric on S2 × S2 with positive flag curvature
K > 0? It is a famous problem in Riemannian geometry to prove the (non)existence of Riemannian metrics on S2 × S2 with positive sectional curvature.
Problem 34 In a Finsler manifold (M, F ), define a parallel translation Pc :
Tp M → Tq M along a curve c(t), a ≤ t ≤ b, by
Pc(V (a)) := V (b),
∂
where V (t) = V i (t) ∂x
i |c(t) is a parallel vector field along c defined by
D̃ċV (t) :=
n dV i
dt
(t) +
o ∂
∂Gi ċ(t),
V
(t)
|c(t) = 0.
∂yj
∂xi
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 Up of p in which any pair of unit speed geodesics γ(t), σ(t),
0 ≤ t ≤ , with γ(0) = σ(0) = p satisfy
1 d γ(s), σ(t) ≤ d γ(2s), σ(2t) ,
2
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0 ≤ s, t ≤ .
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 ⊂ Tx 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 Rn 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 Hij = (n + 1)γF −1 hij or H =
(n2 − 1)γF −1 , where γ is a 1-form.
Hij := Eij|mym ,
H := gij Hij .
Problem 40 Characterize Randers metrics with Hij = 0 or H = 0.
Problem 41 Study Finsler metrics of scalar flag curvature with
3θ
+ σ,
F
where θ is a 1-form and σ is a scalar function on the manifold.
K=
Problem 42 A Finsler metric F on a manifold M is called a W-quadratic
metric if the coefficients of the Weyl curvature W ik = Wj ikl (x)yj yl 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 Wquadratic metrics. Characterize (α, β)-metrics which are R-quadratic or Wquadratic metrics.
Problem 43 Study and characterize the solutions H = H(x, y) = P (x, y) +
iF (x, y) to the following equations on U × Rn :
Hxk = HHyk ,
H(x, λy) = λH(x, y),
λ>0
n
such that F (xo, y) is a Minkowski norm on Txo U = R . Find some solutions
expressed in terms of elementary functions.
Let H = H(x, y) be a function on U × Rn satisfy the following algebraic
equation
zH 2 = |y|2 − 2hy, xiH + |x|2H 2,
where z = a + ib is a non-zero complex number, | · | denotes the standard
Euclidean norm and h , i its inner product. Then for x close to the origin, any
solution H = H(x, y) satisfies
Hxk = HHyk .
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Problem 44 Study Finsler metrics with constant principal curvatures. The
principal curvatures are the eigenvalues of the Riemann curvature Ry : Tx M →
Tx 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 T M with K(x, λy) =
K(x, y), ∀λ > 0, does there exist a Finsler metric whose flag curvature is K?
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