Generalization of Riemannian manifolds
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold "M" where a (possibly asymmetric) Minkowski norm "F"("x", −) is provided on each tangent space T"x""M", that enables one to define the length of any smooth curve "γ" : ["a", "b"] → "M" as
formula_1
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 .
Definition.
A Finsler manifold is a differentiable manifold "M" together with a Finsler metric, which is a continuous nonnegative function "F": T"M" → [0, +∞) defined on the tangent bundle so that for each point "x" of "M",
 λ"F"("v") for all λ ≥ 0 (but not necessarily for λ ; 0) (positive homogeneity).
 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:
The subadditivity axiom may then be replaced by the following strong convexity condition:
Here the Hessian of "F"2 at "v" is the symmetric bilinear form
formula_2
also known as the fundamental tensor of "F" at "v". Strong convexity of "F" implies the subadditivity with a strict inequality if . 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") for all tangent vectors "v".
A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.
Examples.
Randers manifolds.
Let formula_3 be a Riemannian manifold and "b" a differential one-form on "M" with
formula_4
where formula_5 is the inverse matrix of formula_6 and the Einstein notation is used. Then
formula_7
defines a Randers metric on "M" and formula_8 is a Randers manifold, a special case of a non-reversible Finsler manifold.
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:
Then one can define a Finsler function "F": "TM" →[0, ∞] by
formula_10
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 "d""L": "M" × "M" → [0, ∞] of the original quasimetric can be recovered from
formula_11
and in fact any Finsler function "F": T"M" → [0, ∞) defines an intrinsic quasimetric "d""L" on "M" by this formula.
Geodesics.
Due to the homogeneity of "F" the length
formula_12
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
formula_13
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 T"M" as
formula_14
where "k" = 1, ..., "n" and "g"ij is the coordinate representation of the fundamental tensor, defined as
formula_15
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"] → T"M"&amp;setminus;{0} is an integral curve of the smooth vector field "H" on T"M"&amp;setminus;{0} locally defined by
formula_16
where the local spray coefficients "G"i are given by
formula_17
The vector field "H" on T"M"&amp;setminus;{0} satisfies "JH" = "V" and ["V", "H"] = "H", where "J" and "V" are the canonical endomorphism and the canonical vector field on T"M"&amp;setminus;{0}. Hence, by definition, "H" is a spray on "M". The spray "H" defines a nonlinear connection on the fibre bundle T"M"&amp;setminus;{0} → "M" through the vertical projection
formula_18
In analogy with the Riemannian case, there is a version
formula_19
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") ∈ T"M"&amp;setminus;{0} by the uniqueness of integral curves.
If "F"2 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.
Notes.
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