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In [[Mathematics|mathematics]], particularly [[Differential geometry|differential geometry]], a '''Finsler manifold''' is a [[Differentiable manifold|differentiable manifold]] {{math|''M''}} where a (possibly [[Asymmetric norm|asymmetric]]) '''[[Minkowski functional|Minkowski norm]]''' {{math|''F''(''x'', −)}} is provided on each tangent space {{math|T<sub>''x''</sub>''M''}}, that enables one to define the length of any smooth curve {{math|''γ'' : [''a'', ''b''] → ''M''}} as

:<math>L(\gamma) = \int_a^b F\left(\gamma(t), \dot{\gamma}(t)\right)\,\mathrm{d}t.</math>

Finsler manifolds are more general than [[Riemannian manifold]]s since the tangent norms need not be induced by inner products.

Every Finsler manifold becomes an [[Intrinsic metric|intrinsic]] quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.

{{harvs|txt|last=Cartan|first=Élie|year1=1933}} named Finsler manifolds after [[Biography:Paul Finsler|Paul Finsler]], who studied this geometry in his dissertation {{harv|Finsler|1918}}.

==Definition==
A '''Finsler manifold''' is a [[Differentiable manifold|differentiable manifold]] {{math|''M''}} together with a '''Finsler metric''', which is a continuous nonnegative function {{math|''F'': T''M'' → [0, +∞)}} defined on the [[Tangent bundle|tangent bundle]] so that for each point {{math|''x''}} of {{math|''M''}},

* {{math|''F''(''v'' + ''w'') ≤ ''F''(''v'') + ''F''(''w'')}} for every two vectors  {{math|''v'',''w''}} tangent to {{math|''M''}} at {{math|''x''}} ([[Subadditivity|subadditivity]]).
* {{math|''F''(λ''v'') {{=}} λ''F''(''v'')}} for all {{math|λ ≥ 0}} (but not necessarily for&nbsp;{{math|λ < 0)}} ([[Homogeneous function|positive homogeneity]]).
* {{math|''F''(''v'') > 0}} unless {{math|''v'' {{=}} 0}} ([[Positive-definite function|positive definiteness]]).

In other words, {{math|''F''(''x'', −)}} is an [[Asymmetric norm|asymmetric norm]] on each tangent space {{math|T<sub>''x''</sub>''M''}}. The Finsler metric {{math|''F''}} is also required to be '''smooth''', more precisely:

* {{math|''F''}} is smooth on the complement of the zero section of {{math|T''M''}}.

The subadditivity axiom may then be replaced by the following '''strong convexity condition''':

* For each tangent vector {{math|''v'' ≠ 0}}, the [[Hessian matrix]] of {{math|''F''<sup>2</sup>}} at {{math|''v''}} is [[Positive-definite matrix|positive definite]].

Here the Hessian of {{math|''F''<sup>2</sup>}} at {{math|''v''}} is the [[Symmetric tensor|symmetric]] [[Bilinear form|bilinear form]]

:<math>\mathbf{g}_v(X, Y) := \frac{1}{2}\left.\frac{\partial^2}{\partial s\partial t}\left[F(v + sX + tY)^2\right]\right|_{s=t=0},</math>

also known as the '''fundamental tensor''' of {{math|''F''}} at {{math|''v''}}. Strong convexity of {{math|''F''}} implies the subadditivity with a strict inequality if {{math|{{frac|''u''|''F''(''u'')}} ≠ {{frac|''v''|''F''(''v'')}}}}. If {{math|''F''}} is strongly convex, then it is a '''Minkowski norm''' on each tangent space.

A Finsler metric is '''reversible''' if, in addition,

* {{math|''F''(−''v'') {{=}} ''F''(''v'')}} for all tangent vectors ''v''.

A reversible Finsler metric defines a [[Norm (mathematics)|norm]] (in the usual sense) on each tangent space.

==Examples==
* Smooth submanifolds (including open subsets) of a [[Normed vector space|normed vector space]] of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.
* [[Riemannian manifold]]s (but not [[Pseudo-Riemannian manifold|pseudo-Riemannian manifold]]s) are special cases of Finsler manifolds.

===Randers manifolds===
Let <math>(M, a)</math> be a [[Riemannian manifold]] and ''b'' a [[Differential form|differential one-form]] on ''M'' with
:<math>\|b\|_a := \sqrt{a^{ij}b_i b_j} < 1,</math>

where <math>\left(a^{ij}\right)</math> is the inverse matrix of <math>(a_{ij})</math> and the [[Einstein notation]] is used. Then
:<math>F(x, v) := \sqrt{a_{ij}(x)v^i v^j} + b_i(x)v^i</math>

defines a '''Randers metric''' on ''M'' and <math>(M, F)</math> is a '''Randers manifold''', a special case of a non-reversible Finsler manifold.<ref>{{cite journal |first=G. |last=Randers |year=1941 |title=On an Asymmetrical Metric in the Four-Space of General Relativity |journal=[[Physics:Physical Review|Phys. Rev.]] |volume=59 |issue=2 |pages=195–199 |doi=10.1103/PhysRev.59.195 |hdl=10338.dmlcz/134230 |hdl-access=free }}</ref><!--===Kropina manifolds===
'''F'''^3 = (''M'', ''M''', ''F'') is a Kropina manifold (after V. K. Kropina (1959)).-->

===Smooth quasimetric spaces===
Let (''M'', ''d'') be a quasimetric so that ''M'' is also a [[Differentiable manifold|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'',&nbsp;φ) of ''M'' and a constant ''C''&nbsp;≥&nbsp;1 such that for every ''x'',&nbsp;''y''&nbsp;∈&nbsp;''U''
*: <math> \frac{1}{C}\|\phi(y) - \phi(x)\| \leq d(x, y) \leq C\|\phi(y) - \phi(x)\|.</math>
* The function ''d'':&nbsp;''M''&nbsp;×&nbsp;''M''&nbsp;→&nbsp;[0,&nbsp;∞] is smooth in some punctured neighborhood of the diagonal.

Then one can define a Finsler function ''F'':&nbsp;''TM''&nbsp;→[0,&nbsp;∞] by

:<math>F(x, v) := \lim_{t \to 0+} \frac{d(\gamma(0), \gamma(t))}{t},</math>

where ''γ'' is any curve in ''M'' with ''γ''(0)&nbsp;=&nbsp;''x'' and ''γ'''(0)&nbsp;=&nbsp;v. The Finsler function ''F'' obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of ''M''. The [[Intrinsic metric|induced intrinsic metric]] {{nowrap|''d''<sub>''L''</sub>: ''M'' × ''M'' → [0, ∞]}} of the original quasimetric can be recovered from

:<math>d_L(x, y) := \inf\left\{\ \left.\int_0^1 F\left(\gamma(t), \dot\gamma(t)\right) \, dt \ \right| \ \gamma\in C^1([0, 1], M) \ , \ \gamma(0) = x \ , \ \gamma(1) = y \ \right\},</math>

and in fact any Finsler function ''F'':&nbsp;T''M''&nbsp;→&nbsp;<nowiki>[</nowiki>0,&nbsp;∞) defines an [[Intrinsic metric|intrinsic]] quasimetric ''d''<sub>''L''</sub> on ''M'' by this formula.

==Geodesics==
Due to the homogeneity of ''F'' the length

:<math>L[\gamma] := \int_a^b F\left(\gamma(t), \dot{\gamma}(t)\right)\, dt</math>

of a [[Differentiable curve|differentiable curve]] ''γ'': [''a'', ''b''] → ''M'' in ''M'' is invariant under positively oriented [[Parametrization (geometry)|reparametrizations]]. A constant speed curve ''γ'' is a [[Geodesic|geodesic]] of a Finsler manifold if its short enough segments ''γ''|<sub>[''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

:<math>E[\gamma] := \frac{1}{2}\int_a^b F^2\left(\gamma(t), \dot{\gamma}(t)\right)\, dt</math>

in the sense that its [[Functional derivative|functional derivative]] vanishes among differentiable curves {{nowrap|''γ'': [''a'', ''b''] &rarr; ''M''}} with fixed endpoints {{nowrap|''γ''(''a'') {{=}} ''x''}} and {{nowrap|''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''<sup>1</sup>, ..., ''x''<sup>n</sup>, ''v''<sup>1</sup>, ..., ''v''<sup>n</sup>) of T''M'' as

:<math>
  g_{ik}\Big(\gamma(t), \dot\gamma(t)\Big)\ddot\gamma^i(t) + \left(
               \frac{\partial g_{ik}}{\partial x^j}\Big(\gamma(t), \dot\gamma(t)\Big) -
    \frac{1}{2}\frac{\partial g_{ij}}{\partial x^k}\Big(\gamma(t), \dot\gamma(t)\Big)
  \right) \dot\gamma^i(t)\dot\gamma^j(t) = 0,
</math>

where ''k'' = 1, ..., ''n'' and ''g''<sub>ij</sub> is the coordinate representation of the fundamental tensor, defined as

:<math>
g_{ij}(x,v) := g_v\left(\left.\frac{\partial}{\partial x^i}\right|_x, \left.\frac{\partial}{\partial x^j}\right|_x\right).
</math>

Assuming the [[Convex function#Strongly convex functions|strong convexity]] of ''F''<sup>2</sup>(''x'', ''v'') with respect to ''v'' &isin; T<sub>''x''</sub>''M'', the matrix ''g''<sub>''ij''</sub>(''x'', ''v'') is invertible and its inverse is denoted by ''g''<sup>''ij''</sup>(''x'', ''v''). Then {{nobreak|''γ'': [''a'', ''b''] &rarr; ''M''}} is a geodesic of (''M'', ''F'') if and only if its tangent curve {{nobreak|''γ''': [''a'', ''b''] → T''M''&setminus;{0} }} is an [[Integral curve|integral curve]] of the [[Vector field|smooth vector field]] ''H'' on T''M''&setminus;{0} locally defined by

:<math>
\left.H\right|_{(x, v)} := \left.v^i\frac{\partial}{\partial x^i}\right|_{(x,v)}\!\! - \left.2G^i(x, v)\frac{\partial}{\partial v^i}\right|_{(x,v)},
</math>

where the local spray coefficients ''G''<sup>i</sup> are given by

:<math>
G^i(x, v) := \frac{1}{4}g^{ij}(x, v)\left(2\frac{\partial g_{jk}}{\partial x^\ell}(x, v) - \frac{\partial g_{k\ell}}{\partial x^j}(x, v)\right)v^k v^\ell.
</math>

The vector field ''H'' on T''M''&setminus;{0} satisfies ''JH''&nbsp;=&nbsp;''V'' and [''V'',&nbsp;''H'']&nbsp;=&nbsp;''H'', where ''J'' and ''V'' are the [[Double tangent bundle#Canonical tensor fields on the tangent bundle|canonical endomorphism]] and the [[Double tangent bundle#Canonical tensor fields on the tangent bundle|canonical vector field]] on T''M''&setminus;{0}. Hence, by definition, ''H'' is a [[Spray (mathematics)|spray]] on&nbsp;''M''. The spray ''H'' defines a [[Ehresmann connection|nonlinear connection]] on the fibre bundle {{nowrap|T''M''&setminus;{0} → ''M''}} through the [[Ehresmann connection|vertical projection]]

:<math>v: T(\mathrm{T}M \setminus \{0\}) \to T(\mathrm{T}M \setminus \{0\});\quad v := \frac{1}{2}\big(I + \mathcal{L}_H J\big).</math>

In analogy with the [[Riemannian manifold|Riemannian]] case, there is a version

:<math>D_{\dot\gamma}D_{\dot\gamma}X(t) + R_{\dot\gamma}\left(\dot\gamma(t), X(t)\right) = 0</math>

of the Jacobi equation for a general spray structure (''M'', ''H'') in terms of the [[Ehresmann connection|Ehresmann curvature]] and [[Double tangent bundle#Nonlinear covariant derivatives on smooth manifolds|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'',&nbsp;''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''<sup>2</sup> there exists a unique maximal geodesic ''γ'' with ''γ''(0)&nbsp;=&nbsp;x and ''γ'''(0)&nbsp;=&nbsp;v for any (''x'',&nbsp;''v'')&nbsp;∈&nbsp;T''M''&setminus;{0} by the uniqueness of [[Integral curve|integral curve]]s.

If ''F''<sup>2</sup> is strongly convex, geodesics ''γ'':&nbsp;[0,&nbsp;''b'']&nbsp;→&nbsp;''M'' are length-minimizing among nearby curves until the first point ''γ''(''s'') conjugate to ''γ''(0) along ''γ'', and for ''t''&nbsp;>&nbsp;''s'' there always exist shorter curves from ''γ''(0) to ''γ''(''t'') near ''γ'', as in the [[Riemannian manifold|Riemannian]] case.

==Notes==
{{reflist}}

== See also ==
* {{annotated link|Banach manifold}}
* {{annotated link|Fréchet manifold}}
* [[Global analysis]] – which uses Hilbert manifolds and other kinds of infinite-dimensional manifolds
* {{annotated link|Hilbert manifold}}

==References==

*{{Citation | editor1-last=Antonelli | editor1-first=Peter L.|editor1-link=Peter L. Antonelli | title=Handbook of Finsler geometry. Vol. 1, 2 | url=https://books.google.com/books?id=b2B5_IUvPJgC | publisher=Kluwer Academic Publishers | location=Boston | isbn=978-1-4020-1557-1 | mr=2067663 | year=2003}}
* {{cite book|first1=David|last1= Bao| first2=Shiing-Shen |last2=Chern| first3= Zhongmin|last3=Shen|title=An introduction to Riemann–Finsler geometry|series=Graduate Texts in Mathematics|volume= 200|publisher= Springer-Verlag|location=New York|year= 2000|isbn=0-387-98948-X|mr=1747675|doi=10.1007/978-1-4612-1268-3}}
*{{Citation | last1=Cartan | first1=Élie |  title=Sur les espaces de Finsler | zbl=0006.22501 | year=1933 | journal=C. R. Acad. Sci. Paris  | volume=196 | pages=582–586}}
* {{citation|first=Shiing-Shen |last=Chern|   title=Finsler geometry is just Riemannian geometry without the quadratic restriction|journal= [[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]]|volume= 43 |year=1996|issue=9|pages=959–63|url=https://www.ams.org/notices/199609/chern.pdf|mr=1400859}}
*{{Citation | last1=Finsler | first1=Paul | title=Über Kurven und Flächen in allgemeinen Räumen | publisher=Göttingen | series=Dissertation | jfm=46.1131.02 | year=1918}} (Reprinted by Birkhäuser (1951))
* {{cite book|first=Hanno|last= Rund|title=The differential geometry of Finsler spaces|publisher= Springer-Verlag|series=Die Grundlehren der Mathematischen Wissenschaften|volume=101| location=Berlin–Göttingen–Heidelberg| year= 1959|mr=0105726| doi=10.1007/978-3-642-51610-8|isbn=978-3-642-51612-2}}
* {{cite book|first=Zhongmin|last= Shen|title=Lectures on Finsler geometry|publisher= World Scientific|location=Singapore|year= 2001|isbn=981-02-4531-9|mr=1845637|doi=10.1142/4619}}

==External links==
* {{springer|title=Finsler space, generalized|id=p/f040420}}
* [http://finsler.blogspot.com/ The (New) Finsler Newsletter]


[[Category:Differential geometry]]
[[Category:Riemannian geometry]]
[[Category:Riemannian manifolds]]
[[Category:Smooth manifolds]]

{{Sourceattribution|Finsler manifold}}