In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve as
:$L(\backslash gamma)\; =\; \backslash int\_a^b\; F(\backslash gamma(t),\backslash dot(t))\backslash ,\backslash mathrm\; dt.$
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 .

Definition

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 , * for every two vectors tangent to at (subadditivity). * for all (but not necessarily for (positive homogeneity). * unless (positive definiteness). In other words, is an asymmetric norm on each tangent space . The Finsler metric is also required to be smooth, more precisely: * is smooth on the complement of the zero section of . The subadditivity axiom may then be replaced by the following strong convexity condition: * For each tangent vector , the Hessian matrix of at is positive definite. Here the Hessian of at is the symmetric bilinear form :$\backslash mathbf\_v(X,Y)\; :=\; \backslash frac\backslash left.\backslash frac\backslash left(v\; +\; sX\; +\; tY)^2\backslash right\_,$ 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, * 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 pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

Randers manifolds

Let $(M,a)$ be a Riemannian manifold and ''b'' a differential one-form on ''M'' with :$\backslash |b\backslash |\_a\; :=\; \backslash sqrt\; <\; 1,$ where $(a^)$ is the inverse matrix of $(a\_)$ and the Einstein notation is used. Then :$F(x,v)\; :=\; \backslash sqrt\; +\; b\_i(x)v^i$ defines a Randers metric on ''M'' and $(M,F)$ 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: * 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'' :: $\backslash frac\backslash |\backslash phi(y)-\backslash phi(x)\backslash |\; \backslash leq\; d(x,y)\; \backslash leq\; C\backslash |\backslash phi(y)-\backslash phi(x)\backslash |.$ * The function ''d'' : ''M'' × ''M'' →,∞is smooth in some punctured neighborhood of the diagonal. Then one can define a Finsler function ''F'' : ''TM'' →,∞ by :$F(x,v)\; :=\; \backslash lim\_\; \backslash frac,$ 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 :$d\_L(x,y)\; :=\; \backslash inf\backslash left\backslash ,$ and in fact any Finsler function ''F'' : ''TM'' →intrinsic_[[quasimetric.html" style="text-decoration: none;"class="mw-redirect" title="intrinsic metric">intrinsic [[quasimetric">intrinsic metric">intrinsic [[quasimetric ''d''_{''L''} on ''M'' by this formula.

Geodesics

Due to the homogeneity of ''F'' the length :$L[\backslash gamma]:=\backslash int\_a^b\; F(\backslash gamma(t),\backslash dot(t))\backslash ,\; dt$ of a [[differentiable curve ''γ'':[''a'',''b'']→''M'' in ''M'' is invariant under positively oriented [[Parametrization (geometry)|reparametrizations. A constant speed curve ''γ'' is a geodesic of a Finsler manifold if its short enough segments ''γ''|_{'c'',''d''/sub> are length-minimizing in ''M'' from ''γ''(''c'') to ''γ''(''d''). Equivalently, ''γ'' is a geodesic if it is stationary for the energy functional
:$Egamma=\backslash frac\backslash int\_a^b\; F^2(\backslash gamma(t),\backslash dot(t))\backslash ,\; dt$
in the sense that its functional derivative vanishes among differentiable curves 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
:$g\_\backslash Big(\backslash gamma(t),\backslash dot\backslash gamma(t)\backslash Big)\backslash ddot\backslash gamma^i(t)\; +\; \backslash left(\backslash frac\backslash Big(\backslash gamma(t),\backslash dot\backslash gamma(t)\backslash Big)\; -\; \backslash frac\backslash frac\backslash Big(\backslash gamma(t),\backslash dot\backslash gamma(t)\backslash Big)\; \backslash right)\backslash dot\backslash gamma^i(t)\backslash dot\backslash gamma^j(t)\; =\; 0,$
where ''k''=1,...,''n'' and ''g''ij is the coordinate representation of the fundamental tensor, defined as
:$g\_(x,v)\; :=\; g\_v\backslash left(\backslash tfrac\backslash big|\_x,\backslash tfrac\backslash big|\_x\backslash right).$
Assuming the strong convexity of ''F''2(''x,v'') with respect to ''v''∈''TxM'', the matrix ''g''''ij''(''x'',''v'') is invertible and its inverse is denoted by ''g''''ij''(''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'' \0 locally defined by
:$H|\_\; :=\; v^i\backslash tfrac\backslash big|\_\; -\; \backslash \; 2G^i(x,v)\backslash tfrac\backslash big|\_,$
where the local spray coefficients ''G''i are given by
:$G^i(x,v)\; :=\; \backslash frac\backslash left(2\backslash frac(x,v)\; -\; \backslash frac(x,v)\backslash right)v^k\; v^\backslash ell.$
The vector field ''H'' on ''TM''/0 satisfies ''JH'' = ''V'' and 'V'',''H''nbsp;= ''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 through the vertical projection
:$v:T(TM\backslash setminus\; 0)\backslash to\; T(TM\backslash setminus\; 0)\; \backslash quad\; ;\; \backslash quad\; v\; :=\; \backslash tfrac\backslash big(\; I\; +\; \backslash mathcal\; L\_H\; J\; \backslash big).$
In analogy with the Riemannian case, there is a version
:$D\_D\_X(t)\; +\; R\_(\backslash dot\backslash gamma(t),X(t))\; =\; 0$
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'' : , ''b''nbsp;→ ''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
References
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* (Reprinted by Birkhäuser (1951))
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External links
* {{springer|title=Finsler space, generalized|id=p/f040420
The (New) Finsler Newsletter
Category:Finsler geometry
Category:Smooth manifolds
}

Definition

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 , * for every two vectors tangent to at (subadditivity). * for all (but not necessarily for (positive homogeneity). * unless (positive definiteness). In other words, is an asymmetric norm on each tangent space . The Finsler metric is also required to be smooth, more precisely: * is smooth on the complement of the zero section of . The subadditivity axiom may then be replaced by the following strong convexity condition: * For each tangent vector , the Hessian matrix of at is positive definite. Here the Hessian of at is the symmetric bilinear form :$\backslash mathbf\_v(X,Y)\; :=\; \backslash frac\backslash left.\backslash frac\backslash left(v\; +\; sX\; +\; tY)^2\backslash right\_,$ 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, * 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 pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

Randers manifolds

Let $(M,a)$ be a Riemannian manifold and ''b'' a differential one-form on ''M'' with :$\backslash |b\backslash |\_a\; :=\; \backslash sqrt\; <\; 1,$ where $(a^)$ is the inverse matrix of $(a\_)$ and the Einstein notation is used. Then :$F(x,v)\; :=\; \backslash sqrt\; +\; b\_i(x)v^i$ defines a Randers metric on ''M'' and $(M,F)$ 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: * 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'' :: $\backslash frac\backslash |\backslash phi(y)-\backslash phi(x)\backslash |\; \backslash leq\; d(x,y)\; \backslash leq\; C\backslash |\backslash phi(y)-\backslash phi(x)\backslash |.$ * The function ''d'' : ''M'' × ''M'' →,∞is smooth in some punctured neighborhood of the diagonal. Then one can define a Finsler function ''F'' : ''TM'' →,∞ by :$F(x,v)\; :=\; \backslash lim\_\; \backslash frac,$ 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 :$d\_L(x,y)\; :=\; \backslash inf\backslash left\backslash ,$ and in fact any Finsler function ''F'' : ''TM'' →

Geodesics

Due to the homogeneity of ''F'' the length :$L[\backslash gamma]:=\backslash int\_a^b\; F(\backslash gamma(t),\backslash dot(t))\backslash ,\; dt$ of a [[differentiable curve ''γ'':[''a'',''b'']→''M'' in ''M'' is invariant under positively oriented [[Parametrization (geometry)|reparametrizations. A constant speed curve ''γ'' is a geodesic of a Finsler manifold if its short enough segments ''γ''|