In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski norm is provided on each tangent space, that enables one to define the length of any smooth curve as
L(\gamma)=
| b | |
| \int | |
| a |
F\left(\gamma(t),
| \gamma |
(t)\right)dt.
Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.
Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.
named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation .
A Finsler manifold is a differentiable manifold together with a Finsler metric, which is a continuous nonnegative function defined on the tangent bundle so that for each point of,
In other words, is an asymmetric norm on each tangent space . The Finsler metric is also required to be smooth, more precisely:
The subadditivity axiom may then be replaced by the following strong convexity condition:
Here the Hessian of at is the symmetric bilinear form
gv(X,Y):=
| 1 | \left. | |
| 2 |
| \partial2 | |
| \partials\partialt |
\left[F(v+sX+
| 2\right]\right| | |
| tY) | |
| s=t=0 |
,
also known as the fundamental tensor of at . Strong convexity of implies the subadditivity with a strict inequality if . If is strongly convex, then it is a Minkowski norm on each tangent space.
A Finsler metric is reversible if, in addition,
A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.
Let
(M,a)
\|b\|a:=\sqrt{aijbibj}<1,
where
\left(aij\right)
(aij)
F(x,v):=\sqrt{aij(x)vivj}+
| i | |
| b | |
| i(x)v |
defines a Randers metric on M and
(M,F)
Let (M, d) be a quasimetric so that M is also a differentiable manifold and d is compatible with the differential structure of M in the following sense:
| 1 | |
| C |
\|\phi(y)-\phi(x)\|\leqd(x,y)\leqC\|\phi(y)-\phi(x)\|.
Then one can define a Finsler function F: TM →[0, ∞] by
F(x,v):=\limt
| d(\gamma(0),\gamma(t)) | |
| t |
,
where γ is any curve in M with γ(0) = x and γ(0) = v. The Finsler function F obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of M. The induced intrinsic metric of the original quasimetric can be recovered from
dL(x,y):=
| 1 | |
| inf\left\{ \left.\int | |
| 0 |
F\left(\gamma(t),
| \gamma(t)\right) |
dt \right| \gamma\inC1([0,1],M) , \gamma(0)=x , \gamma(1)=y \right\},
and in fact any Finsler function F: TM → [0, ∞) defines an intrinsic quasimetric dL on M by this formula.
Due to the homogeneity of F the length
L[\gamma]:=
| b | |
| \int | |
| a |
F\left(\gamma(t),
| \gamma |
(t)\right)dt
of a differentiable curve γ: [''a'', ''b''] → M in M is invariant under positively oriented reparametrizations. A constant speed curve γ is a geodesic of a Finsler manifold if its short enough segments γ|[''c'',''d''] are length-minimizing in M from γ(c) to γ(d). Equivalently, γ is a geodesic if it is stationary for the energy functional
E[\gamma]:=
| 1 | |
| 2 |
| b | |
| \int | |
| a |
F2\left(\gamma(t),
| \gamma |
(t)\right)dt
in the sense that its functional derivative vanishes among differentiable curves with fixed endpoints and .
The Euler–Lagrange equation for the energy functional E[''γ''] reads in the local coordinates (x1, ..., xn, v1, ..., vn) of TM as
gik(\gamma(t),
| \gamma(t))\ddot\gamma |
i(t)+\left(
| \partialgik | |
| \partialxj |
(\gamma(t),
| \gamma(t)) |
-
| 1 | |
| 2 |
| \partialgij | |
| \partialxk |
(\gamma(t),
| \gamma(t)) |
\right)
| \gamma |
| |||
j(t)=0,
where k = 1, ..., n and gij is the coordinate representation of the fundamental tensor, defined as
gij(x,v):=
| g | ||||
|
\right|x,\left.
| \partial | |
| \partialxj |
\right|x\right).
Assuming the strong convexity of F2(x, v) with respect to v ∈ TxM, the matrix gij(x, v) is invertible and its inverse is denoted by gij(x, v). Then is a geodesic of (M, F) if and only if its tangent curve is an integral curve of the smooth vector field H on TM∖ locally defined by
\left.H\right|(x,:=
| ||||
| \left.v |
\right|(x,v)-\left.2Gi(x,v)
| \partial | |
| \partialvi |
\right|(x,v),
where the local spray coefficients Gi are given by
Gi(x,v):=
| 1 | |
| 4 |
gij(x,v)\left(2
| \partialgjk | |
| \partialx\ell |
(x,v)-
| \partialgk\ell | |
| \partialxj |
(x,v)\right)vkv\ell.
The vector field H on TM∖ satisfies JH = V and [''V'', ''H''] = H, where J and V are the canonical endomorphism and the canonical vector field on TM∖. Hence, by definition, H is a spray on M. The spray H defines a nonlinear connection on the fibre bundle through the vertical projection
v:T(TM\setminus\{0\})\toT(TM\setminus\{0\}); v:=
| 1 | |
| 2 |
(I+l{L}HJ).
In analogy with the Riemannian case, there is a version
| D | |||
|
| D | |||
|
X(t)+
| R | \left( | |||
|
| \gamma(t), |
X(t)\right)=0
of the Jacobi equation for a general spray structure (M, H) in terms of the Ehresmann curvature and nonlinear covariant derivative.
By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough neighborhoods) on (M, F). Length minimizing curves can always be positively reparametrized to be geodesics, and any geodesic must satisfy the Euler–Lagrange equation for E[''γ'']. Assuming the strong convexity of F2 there exists a unique maximal geodesic γ with γ(0) = x and γ(0) = v for any (x, v) ∈ TM∖ by the uniqueness of integral curves.
If F2 is strongly convex, geodesics γ: [0, ''b''] → M are length-minimizing among nearby curves until the first point γ(s) conjugate to γ(0) along γ, and for t > s there always exist shorter curves from γ(0) to γ(t) near γ, as in the Riemannian case.
. Hanno Rund. The differential geometry of Finsler spaces. Springer-Verlag. Die Grundlehren der Mathematischen Wissenschaften. 101. Berlin–Göttingen–Heidelberg. 1959. 0105726. 10.1007/978-3-642-51610-8. 978-3-642-51612-2.