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28

ROBERT L
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Structures satisfying K = 1
In this section, I want to study the (generalized) Finsler structures that satisfy K = 1 and construct examples of such on the standard 2-sphere
...
In this case, the structure equations become
dω =

η∧θ

dθ = −η ∧ ω + S θ
dη = − ω + C η

∧θ

dS = C ω + S2 θ + S3 η
dC = −S ω + C2 θ + C3 η
Let W, T, and H denote as usual the dual vector fields to the coframing ω, θ, η
...
e
...
More interesting than
this function, however, are the 1-form S θ + C η, the 2-form η ∧θ, and the quadratic
form η 2 + θ2
...

In fact, LW ω = 0, LW θ = η, and LW η = −θ, which implies
exp∗ ω = ω
tW
exp∗ θ =
tW
exp∗
tW

cos t θ + sin t η

η = − sin t θ + cos t η

These formulae have some interesting consequences for geodesically complete
structures with K ≡ 1
...
Then Φ : Σ → Σ is a
symmetry of the generalized Finsler structure
...
Since Φk is a symmetry of the coframing and Σ is connected, this
implies that Φk is the identity map
...
8
Another interesting observation is that, if Σ is compact, then so is the group G
of symmetries the generalized Finsler structure
...
Now, it
can be shown that there cannot be a compact example with a 2-dimensional abelian
subgroup of the group of symmetries, so the only possibilities for a compact example with positive dimensional symmetry group are the quotients of the standard
Riemannian structure on the 3-sphere and examples with a 1-parameter symmetry
group
...

8 Actually,

if any geodesic on which J is non-zero closes, then it must have length an integral
multiple of 2π
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FINSLER SURFACES

29

5
...
Canonical structures on the geodesic space
...
e
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Then there exist a
1-form ϕ on Λ so that ∗ ϕ = −S θ−C η; a 2-form dA on Λ so that ∗ (dA) = η ∧θ; and
a positive definite quadratic form g on Λ so that ∗ g = η 2 + θ2
...

5
...
Recovering the generalized Finsler structure
...
To see
this, let ν : F → Λ denote the oriented orthonormal frame bundle of Λ with respect
to the metric g and orientation form dA
...

(The facts that (η ∧θ)(H, T) ≡ 1 and that H and T are the duals to η and θ ensures
that ∗ (Hu ), ∗ (Tu ) is a dA-oriented, g-orthonormal frame at (u) ∈ Λ
...
The defining equations of the Levi-Civita connection 1-form α21
on F are
dα2 = −α21 ∧ α1
dα1 = α21 ∧ α2
so pulling these back via ˆ yields
dη = ˆ∗ (α21 ) ∧ θ

dθ = − ˆ∗ (α21 ) ∧ η
...


In particular, ˆ∗ α1 ∧α2 ∧α21 = −η ∧θ∧ω = 0, so ˆ is a local diffeomorphism
...
In particular, computing the exterior derivative of the
equation ˆ∗(−α21 + ν ∗ ϕ) = ω yields
∗

(dA) = η ∧ θ = dω = ˆ∗ −dα21 + ν ∗ (dϕ) = ˆ∗ ν ∗ R dA + dϕ =

∗

R dA + dϕ ,

so that dϕ = (1−R) dA
...
Then on the orthonormal frame bundle ν : F → Λ with tautological
1-forms α1 and α2 and Levi-Civita connection 1-form α21 satisfying the standard
structure equations
dα1 = α21 ∧ α2
dα2 = −α21 ∧ α1
dα21 = −ν ∗ R α1 ∧ α2

30

ROBERT L
...
Thus, the given
data (g, dA, ϕ) on the surface Λ suffice to determine a local solution to the K = 1
equation
...
2
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A normal form
...
However, this last step can also
be avoided in a more-or-less natural way
...
There is a natural map from the
¯ ¯
oriented orthonormal frame bundle F of g to F that simply scales a g-orthonormal
¯
−u
¯ → F
...
I will denote this map by µ : F
yield
¯
µ∗ α1 = eu α1
µ∗ α2 = eu α2
¯
µ∗ α21 = α21 + ∗du
¯
Pulling back the equation dα21 = ν ∗ dϕ − dA via µ then yields
¯
¯
d α21 + ∗du = ν ∗ dϕ − e2u dA ,
¯
so setting ϕ = ϕ − ∗du, this can be written
¯
¯
dϕ + R dA = e2u dA
...
Then define
¯
and choose any 1-form ϕ satisfying the open condition dϕ + R
¯
¯
¯
the function u by the equation dϕ + R dA = e2u dA
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FINSLER SURFACES

31

¯
Explicitly, in the case R = 0, let x and y be local coordinates on V ⊂ Λ so that,
2
¯
¯
on V , one has g = dx + dy 2 and dA = dx∧dy
...
Then on V × S 1 FV , the 1-forms
ω = −dφ + a dx + b dy
¯
¯
θ = bx −ay − sin φ dx + cos φ dy
η=
¯

bx −ay

cos φ dx + sin φ dy

satisfy the structure equations of a generalized Finsler structure with K = 1 and
every generalized Finsler structure with K = 1 is locally of this form
...

5
...
Compact examples
...

5
...
1
...
The case where S (and hence, C) vanishes identically is the (generalized) Riemannian case, for which there is a unique local model
...
Thus Σ = Γ\ SU(2)
where Γ ⊂ SU(2) is a finite subgroup
...
3
...
The non-Riemannian case
...

Since dS ≡ Cω mod {θ, η} and dC ≡ −Sω mod {θ, η}, it follows that S cannot
be constant
...

In particular, the dimension of this group must be at most 2
...

The first problem is how to describe the actual Finsler structures with K = 1
...

To see what the conditions should be, first suppose that Σ ⊂ T M is an actual
Finsler structure satisfying K = 1 and let ω, θ, and η denote the canonical 1-forms
satisfying the structure equations as above
...

Thus, an obvious necessary condition for a generalized Finsler structure to be a
Finsler structure is that all the leaves of the system ω = θ = 0 be compact
...

The first difficulty is that even though all of the leaves may be compact, so
that the system ω = θ = 0 defines a foliation by circles, there may exist certain
exceptional circles (necessarily isolated) around which the foliation is not locally
product of a circle and a 2-disk
...
One can deal with this either by
considering Finsler structures on orbifolds, which does not significantly change the
local theory, or by simply deleting the offending circles, yielding a smooth quotient
space
...
BRYANT

The second difficulty is that even if the foliation is locally a product, with leaf
space quotient π : Σ → M , the map ι : Σ → T M may immerse each fiber Σx
into Tx M as a convex curve which winds around the origin more than once, say m >
1 times
...

However, there is no reason a priori for the images to be simple closed curves, even
in the case where K = 1, as will be seen
...
3
...
The local non-Riemannian case
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(Such disks exist, for example, the geodesic ball Br (x) satisfies this condition for sufficiently small r
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Conversely, for any two distinct points p− and p+ in ∂D,
there is a unique geodesic segment γ lying in D and starting at p− and ending
at p+
...
Let ΣD = π −1 (D)
...
This yields the double fibration
ΣD
ΛD

π
D
...

Now, assuming K ≡ 1 for the original Finsler structure, let g, dA and ϕ be the
canonical metric, area form, and 1-form, respectively, constructed earlier on ΛD
...

The geometric interpretation of these integral curves is as follows: For any unit
speed (oriented) curve γ : [a, b] → ΛD , there are two natural functions associated
with it
...
I will say that γ is a ϕ-geodesic
if κ = f
...
9 Note that the λ-geodesics are
solutions of a second order ODE and that there is exactly one oriented, unit-speed
λ-geodesic in each tangent direction through each point of ΛD
...
Moreover, each closed ϕgeodesic C is the projection of a unique closed integral curve of the system α2 =
α21 − ν ∗ ϕ = 0 which must therefore constitute a single π-fiber, say π −1 (x)
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FINSLER SURFACES

33

generator of the fundamental group of ΛD
...

Thus, each point of ΛD has an open interval of tangent directions for which the
corresponding ϕ-geodesic is closed and represents a generator of the fundamental
group of ΛD
...
Since ΛD is an annulus, the Uniformization Theorem implies that
ΛD , g is conformally equivalent to a standard annulus
...
By the previous local
discussion, I can then embed ΣD as an open subset of I ×S 1 ×S 1 (with coordinates
(r, ψ, φ)) in such a way that, for some functions a and b of r and ψ with (br −aψ ) > 0
the following formulae hold,
ω = − dφ + a dr + b dψ
θ=

br −aψ − sin φ dr + cos φ dψ

η=

br −aψ

cos φ dr + sin φ dψ

The functions a and b must then be chosen so that an open subset of the integral
curves of the system θ = ω = 0 are closed and project to I ×S 1 to become generators
of the fundamental group
...
3
...
Rotationally invariant examples
...
10 The simplest way to proceed to these examples
is to try to find ones which are rotationally symmetric with respect to the cyclic
coordinate ψ, i
...
, so that aψ = bψ = 0
...
A computation shows that the
invariant S is given by
S=−
10 Such

b (r) + 2b(r)b (r)
2 b (r)

3/2

cos φ
...
The interested reader might want to consult Besse’s
treatise [Be] for further developments of these ideas, both for surfaces and for higher dimensional
manifolds
...
BRYANT

Thus, as long as b + b2 is not constant, the examples constructed will not be
Riemannian
...
To see this, note that
b(r) − sin φ dr + cos φ dψ − cos φ −dφ + b(r) dψ = d sin φ − sin φ b(r) dr
= eB d e−B sin φ
...
In particular, the zero level surface is foliated by the curves defined
by ψ ≡ ψ0 and φ ≡ 0 or π
...
Since the system is invariant under the involution (r, ψ, φ) →
(r, −ψ, −φ), it suffices to look at the positive level surfaces of f
...
Then B has a smooth
inverse on I
...
The equation e−B sin φ = e−B0 can be solved on C for r in the form
r = B −1 eB0 sin φ
...
Moreover, I
claim that φ must be monotone on C, which implies that it cannot be closed
...
However, since b is non-vanishing
and ω = −dφ + b dψ, it follows that neither dφ nor dψ can vanish
...

Thus, from now on, I assume that b does vanish somewhere on I
...
By translation in the variable r, I can assume that b(0) =
0 and can also make B unique by setting B(0) = 0
...
Note that ρ satisfies −π/2 < ρ < π/2
...
I am now going to use ρ as the
natural parameter on I, so that I can regard r as a function of ρ, instead of the
other way around and write r for dr/dρ
...
The level set f = 1 is just ρ = 0
and φ = π/2, which is a closed integral curve whose projection into I × S 1 is a
generator for the fundamental group
...

In particular, since cos ρ and sin φ cannot vanish on Cρ0 , it follows that on any
integral curve of the system which lies in f = cos ρ0 , the equation can be written
in the form
sin ρ dψ
cos φ dψ
dφ =
,
dρ =
r sin φ
˙
r cos ρ
˙
so that dψ cannot vanish on any such integral curve, so I can regard it as a parameter
on the curve
...
Thus, the integral curve will

FINSLER SURFACES

35

close if and only if ψ increases by an rational multiple of π when this point makes
a complete circuit around Cρ0
...


The condition on r needed to satisfy the π1 -generation condition is that P (ρ0 ) ≡ 2π
for all ρ0 in the range 0 < ρ0 < π/2
...


It follows that the necessary and sufficient condition that P (ρ0 ) ≡ 2π for all ρ0 in
˙
˙
the range 0 < ρ0 < π/2 is that r be of the form r = 1 + h(ρ) /(cos ρ) where h is
an odd function defined on an interval symmetric about ρ = 0
...

˙
The identity b dr = b r dρ = −eB d e−B ) = cot ρ dρ yields b = sin ρ/ 1 + h(ρ) and
˙
the condition br = db/dr > 0 becomes
1 + h(ρ) − h (ρ) tan ρ > 0,
which I also impose
...

5
...
5
...
Now, I further require that the odd function h be
defined and smooth on all of R, that it satisfy h(ρ + π) = −h(ρ) and that h
and h vanish at ρ = ±π/2
...
By the general procedure
described above, as long as h is non-zero, this will induce a non-Riemannian Finsler
structure satisfying K ≡ 1 on the 2-sphere of oriented ϕ-geodesics
...
)
6
...

Thus, let ω, θ, η be a generalized Finsler structure on a connected 3-manifold Σ
that satisfies K ≡ −1
...
BRYANT

These structure equations already have global implications
...
Then it would necessarily be geodesically complete,
but the structure equations above show that, along any integral curve of W, the
function S satisfies an equation of the form S − S = 0, where the prime denoted
differentiation with respect to the flow parameter along the integral curve
...
Since Σ is supposed to be compact, S must be bounded on Σ and
hence on every integral curve
...
In particular any
Finsler structure on a compact surface M which satisfies K ≡ −1 (or any negative
constant, for that matter) must be a Riemannian metric11 , a result to be found
in [Ak]
...
e
...
Thus, in the nonRiemannian case, the geodesic flow must be completely integrable, again, a great
contrast with the Riemannian case when K = −1
...
1
...
Consider the 1-form S θ −
C η, the 2-form η ∧θ, and the quadratic form η 2 − θ2
...

If one assumes that the generalized Finsler structure is geodesically amenable,
with geodesic projection : Σ → Λ, then it follows that there exist on Λ a 1-form ϕ
so that ∗ ϕ = −S θ + C η; a 2-form dA so that ∗ dA = η ∧θ; and a Lorentzian
quadratic form g so that ∗g = η 2 − θ2
...
The curvature R of this metric is then defined by dψ = R η ∧θ
and is well-defined on Λ
...

as the equation relating the 1-form ϕ with the oriented Lorentzian structure defined
by g and the choice of oriented area form dA
...
I will now describe this construction
...

Let : Σ → Λ be the bundle of oriented, time oriented g-frames on Λ of the form
(p; e1, e2 ) where p is a point of Λ and (e1 , e2 ) are an oriented, time oriented basis
of TpΛ which satisfies
1 = g(e1 , e1 ) = −g(e2 , e2 ),

0 = g(e1, e2 )
...
It satisfies dψ = ∗ R α1 ∧α2 , where R is the
function on Λ which represents the curvature of g
...
Then the hypothesis that dϕ = (1 − R) dA ensures that the 1-forms
ω=ψ+
θ = α2
η = α1

∗

ϕ = ψ + C α1 − S α2

satisfy the structure equations of a generalized Finsler structure on Σ with K ≡ −1
...

6
...
A local normal form
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Details will be left to the reader
...
3
...
To construct complete examples of actual Finsler
structures on surfaces, it suffices to construct an example of a Λ endowed with
the appropriate structures so that the positive ϕ-geodesics are are all closed and
satisfying appropriate growth conditions
...
The same sort of analysis that produced the compact examples of Finsler
surfaces with K = −1 leads to the following prescription: Let Λ be the cylinder (−π/2, π/2) × S 1 with coordinates ρ and φ (which is periodic of period 2π)
...
Let h : (−π/2, π/2) → (−1, 1)
be an odd function of ρ which satisfies the inequality
1 + h(ρ) − h (ρ) sin ρ cos ρ > 0
...
BRYANT

Now consider the coframing on Σ defined by
ω = dz − (tan ρ)/ 1 + h(ρ) dφ
θ = R(ρ) 1 + h(ρ) cosh z dρ + sinh z dφ
η = R(ρ) 1 + h(ρ) sinh z dρ + cosh z dφ
where the function R > 0 is defined by
R(ρ)

2

=

1 + h(ρ) − h (ρ) sin ρ cos ρ
cos2 ρ 1 + h(ρ)

3


...
A computation shows that the invariant S vanishes
if and only if h satisfies the differential equation
1
1 + h(ρ)

2

−

h (ρ) tan ρ
1 + h(ρ)

3

=c

for some constant c
...
)
It is not hard to show that the ωθ integral curves are closed in Σ for any such
choice of h
...
) They foliate Σ and the leaf projection π : Σ → M induces a Finsler structure on M (topologically a disk) which
satisfies K ≡ −1
...
Thus, these choices of h
provide examples of complete non-Riemannian Finsler structures on the disk that
satisfy K ≡ −1
...
Akbar-Zadeh, Sur les espaces de Finsler a courbures sectionnelles constantes, Acad
...
Belg
...
Cl
...
(5) 74 (1988), 281–322
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Bao, S
...
Chern, and Z
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[BrChG] R
...

[Be]
A
...
Math
...
Band 93,
Springer-Verlag, Berlin and New York, 1978
...
Bryant, Two exotic holonomies in dimension four, path geometries, and twistor
theory, Complex Geometry and Lie Theory, Proceedings of Symposia in Pure Mathematics, vol
...
33–88
...
Bryant, Lectures on the Geometry of Partial Differential Equations, CRM Monograph Series, American Mathematical Society, Providence, 1996 (manuscript in preparation)
...
Cartan, Les Espace M´trique Fond´s sur la Notion d’Aire, Expos´s de G´ometrie,
e
e
e
e
t
...

´
[Ca2]
E
...
79, Hermann, Paris, 1934
...
Cartan, Les espaces g´n´ralis´s et l’int´gration de certaines classes d’´quations
e e
e
e
e
diff´rentielles, C
...
Acad
...
206 (1938), 1689–1693
...
Cartan, La geometria de las ecuaciones diferenciales de tercer orden, Revista
...

Hispano-Amer
...

[Ak]



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