Finsler manifold
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In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm^{[disambiguation needed]} F(x,−) is provided on each tangent space T_{x}M, allowing to define the length of any smooth curve γ : [a,b] → M as
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.
Élie Cartan (1933) named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation (Finsler 1918).
Contents
Definition
A Finsler manifold is a differentiable manifold M together with a Finsler metric, which is a continuous nonnegative function F:TM→[0,+∞) defined on the tangent bundle so that for each point x of M,
 F(v + w) ≤ F(v) + F(w) for every two vectors v,w tangent to M at x (subadditivity).
 F(λv) = λF(v) for all λ ≥ 0 (but not necessarily for λ < 0) (positive homogeneity).
 F(v) > 0 unless v = 0 (positive definiteness).
In other words, F(x,−) is an asymmetric norm on each tangent space T_{x}M. The Finsler metric F is also required to be smooth, more precisely:
 F is smooth on the complement of the zero section of TM.
The subadditivity axiom may then be replaced by the following strong convexity condition:
 For each tangent vector v ≠ 0, the hessian of F^{2} at v is positive definite.
Here the hessian of F^{2} at v is the symmetric bilinear form
also known as the fundamental tensor of F at v. Strong convexity of implies the subadditivity with a strict inequality if ^{u}⁄_{F(u)} ≠ ^{v}⁄_{F(v)}. If F is strongly convex, then it is a Minkowski norm on each tangent space.
A Finsler metric is reversible if, in addition,
 F(−v) = F(v) for all tangent vectors v.
A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.
Examples
 Smooth submanifolds (including open subsets) of a normed vector space of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.
 Riemannian manifolds (but not pseudoRiemannian manifolds) are special cases of Finsler manifolds.
Randers manifolds
Let (M,a) be a Riemannian manifold and b a differential oneform on M with
where is the inverse matrix of and the Einstein notation is used. Then
defines a Randers metric on M and (M,F) is a Randers manifold, a special case of a nonreversible Finsler manifold.^{[1]}
Smooth quasimetric spaces
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:
 Around any point z on M there exists a smooth chart (U, φ) of M and a constant C ≥ 1 such that for every x,y ∈ U
 The function d : M × M →[0,∞] is smooth in some punctured neighborhood of the diagonal.
Then one can define a Finsler function F : TM →[0,∞] by
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 nonMinkowski) norm on each tangent space of M. The induced intrinsic metric d_{L}: M × M → [0, ∞] of the original quasimetric can be recovered from
and in fact any Finsler function F : TM → [0, ∞) defines an intrinsic quasimetric d_{L} on M by this formula.
Geodesics
Due to the homogeneity of F the length
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 lengthminimizing in M from γ(c) to γ(d). Equivalently, γ is a geodesic if it is stationary for the energy functional
in the sense that its functional derivative vanishes among differentiable curves γ:[a,b]→M with fixed endpoints γ(a)=x and γ(b)=y.
Canonical spray structure on a Finsler manifold
The Euler–Lagrange equation for the energy functional E[γ] reads in the local coordinates (x^{1},...,x^{n},v^{1},...,v^{n}) of TM as
where k=1,...,n and g_{ij} is the coordinate representation of the fundamental tensor, defined as
Assuming the strong convexity of F^{2}(x,v) with respect to v∈T_{x}M, the matrix g_{ij}(x,v) is invertible and its inverse is denoted by g^{ij}(x,v). Then γ:[a,b]→M is a geodesic of (M,F) if and only if its tangent curve γ':[a,b]→TM \0 is an integral curve of the smooth vector field H on TM \0 locally defined by
where the local spray coefficients G^{i} are given by
The vector field H on TM/0 satisfies JH = V and [V,H] = H, where J and V are the canonical endomorphism and the canonical vector field on TM \0. Hence, by definition, H is a spray on M. The spray H defines a nonlinear connection on the fibre bundle TM \0 → M through the vertical projection
In analogy with the Riemannian case, there is a version
of the Jacobi equation for a general spray structure (M,H) in terms of the Ehresmann curvature and nonlinear covariant derivative.
Uniqueness and minimizing properties of geodesics
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 F^{2} there exists a unique maximal geodesic γ with γ(0) = x and γ'(0) = v for any (x, v) ∈ TM \ 0 by the uniqueness of integral curves.
If F^{2} is strongly convex, geodesics γ : [0, b] → M are lengthminimizing 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.
Notes
 ^ Randers, G. (1941). "On an Asymmetrical Metric in the FourSpace of General Relativity". Phys. Rev. 59 (2): 195–199. doi:10.1103/PhysRev.59.195.
References
 Antonelli, P. L., ed. (2003), Handbook of Finsler geometry. Vol. 1, 2, Boston: Kluwer Academic Publishers, ISBN 9781402015571, MR 2067663
 D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, SpringerVerlag, 2000. ISBN 038798948X.
 Cartan, Elie (1933), "Sur les espaces de Finsler", C. R. Acad. Sci. Paris, 196: 582–586, Zbl 0006.22501
 S. Chern: Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices AMS, 43 (1996), pp. 959–63.
 Finsler, Paul (1918), Über Kurven und Flächen in allgemeinen Räumen, Dissertation, Göttingen, JFM 46.1131.02 (Reprinted by Birkhäuser (1951))
 H. Rund. The Differential Geometry of Finsler Spaces, SpringerVerlag, 1959. ASIN B0006AWABG.
 Z. Shen, Lectures on Finsler Geometry, World Scientific Publishers, 2001. ISBN 9810245319.
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "Finsler space, generalized", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Z. Shen's Finsler Geometry Website.
 The (New) Finsler Newsletter