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| In [[mathematics]], particularly [[differential geometry]], a '''Finsler manifold''' is a [[differentiable manifold]] together with the structure of an [[intrinsic equation|intrinsic]] [[quasimetric space#Quasimetrics|quasimetric space]] in which the length of any [[rectifiable curve]] {{nowrap|''γ'' : [''a'',''b''] → ''M''}} is given by the length [[functional (mathematics)|functional]]
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| :<math>L[\gamma] = \int_a^b F(\gamma(t),\dot{\gamma}(t))\,dt,</math>
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| where ''F''(''x'', · ) is a '''Minkowski norm''' (or at least an [[asymmetric norm]]) on each [[tangent space]] ''T''<sub>''x''</sub>''M''. Finsler manifolds non-trivially generalize [[Riemannian manifold]]s in the sense that they are not necessarily infinitesimally [[Euclidean space|Euclidean]]. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an [[inner product]] ([[metric tensor]]).
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| {{harvs|txt|authorlink=Élie Cartan|last=Cartan|first=Élie|year1=1933}} named Finsler manifolds after [[Paul Finsler]], who studied this geometry in his dissertation {{harv|Finsler|1918}}.
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| ==Definition==
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| A '''Finsler manifold''' is a [[differentiable manifold]] ''M'' together with a '''Finsler function''' ''F'' defined on the [[tangent bundle]] of ''M'' so that for all tangent vectors ''v'',
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| * ''F'' is [[smooth function|smooth]] on the complement of the zero section of ''TM''.
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| * ''F''(''v'') ≥ 0 with equality if and only if ''v'' = 0 ([[positive-definite function|positive definiteness]]).
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| * ''F''(λ''v'') = λ''F''(''v'') for all λ ≥ 0 (but not necessarily for λ < 0) ([[homogeneous function|homogeneity]]).
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| * ''F''(''v'' + ''w'') ≤ ''F''(''v'') + ''F''(''w'') for all ''w'' at the same tangent space with ''v'' ([[subadditivity]]).
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| In other words, ''F'' is an [[asymmetric norm]] on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition:
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| * For each tangent vector ''v'', the [[Hessian matrix|hessian]] of ''F''<sup>2</sup> at ''v'' is [[Positive-definite matrix|positive definite]].
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| Here the hessian of ''F''<sup>2</sup> at ''v'' is the [[symmetric tensor|symmetric]] [[bilinear form]]
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| :<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>
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| also known as the '''fundamental tensor''' of ''F'' at ''v''. Strong convexity of ''F''<sup>2</sup> implies the subadditivity with a strict inequality if ''u''/''F''(''u'') ≠ ''v''/''F''(''v''). If ''F''<sup>2</sup> is strongly convex, then ''F'' is a '''Minkowski norm''' on each tangent space.
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| A Finsler metric is '''reversible''' if, in addition,
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| * ''F''(−''v'') = ''F''(''v'') for all tangent vectors ''v''.
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| A reversible Finsler metric defines a [[norm (mathematics)|norm]] (in the usual sense) on each tangent space.
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| ==Examples==
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| * [[Normed vector space]]s of finite dimension, such as [[Euclidean space]]s, whose norms are smooth outside the origin.
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| * [[Riemannian manifold]]s (but not [[pseudo-Riemannian manifold]]s) are special cases of Finsler manifolds.
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| ===Randers manifolds===
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| Let (''M'',''a'') be a [[Riemannian manifold]] and ''b'' a [[differential form|differential one-form]] on ''M'' with
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| :<math> \|b\|_a := \sqrt{a^{ij}b_i b_j} < 1,</math>
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| where <math>(a^{ij})</math> is the [[inverse matrix]] of <math>(a_{ij})</math> and the [[Einstein notation]] is used. Then
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| :<math> F(x,v) := \sqrt{a_{ij}(x)v^iv^j} + b_i(x)v^i</math>
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| defines a '''Randers metric''' on ''M'' and (''M'',''F'') 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=[[Physical Review|Phys. Rev.]] |volume=59 |issue=2 |pages=195–199 |doi=10.1103/PhysRev.59.195 }}</ref>
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| <!--===Kropina manifolds===
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| '''F'''^3 = (''M'',''M''',''F'') is a Kropina manifold (after V. K. Kropina (1959)).-->
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| ===Smooth quasimetric spaces===
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| 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:
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| * 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''
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| :: <math> \frac{1}{C}\|\varphi(y)-\varphi(x)\| \leq d(x,y) \leq C\|\varphi(y)-\varphi(x)\|.</math>
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| * The function ''d'' : ''M'' × ''M'' →[0,∞[ is [[smooth function|smooth]] in some punctured neighborhood of the diagonal.
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| Then one can define a Finsler function ''F'' : ''TM'' →[0,∞[ by
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| :<math>F(x,v) := \lim_{t\to 0+} \frac{d(\gamma(0),\gamma(t))}{t},</math>
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| 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 [[intrinsic metric|induced intrinsic metric]] {{nowrap|''d''<sub>''L''</sub>: ''M'' × ''M'' → [0, ∞]}} of the original [[quasimetric]] can be recovered from
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| :<math> d_L(x,y) := \inf\left\{\ \left.\int_0^1 F(\gamma(t),\dot\gamma(t)) \, dt \ \right| \ \gamma\in C^1([0,1],M) \ , \ \gamma(0)=x \ , \ \gamma(1)=y \ \right\},</math>
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| and in fact any Finsler function ''F'' : ''TM'' → <nowiki>[</nowiki>0, ∞<nowiki>)</nowiki> defines an [[intrinsic metric|intrinsic]] [[quasimetric]] ''d''<sub>''L''</sub> on ''M'' by this formula.
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| ==Geodesics==
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| Due to the homogeneity of ''F'' the length
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| :<math>L[\gamma]:=\int_a^b F(\gamma(t),\dot{\gamma}(t))\, dt</math>
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| of a [[differentiable curve]] ''γ'':[''a'',''b'']→''M'' in ''M'' is invariant under positively oriented [[Parametrization|reparametrizations]]. A constant speed curve ''γ'' is a [[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
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| :<math>E[\gamma]:=\frac{1}{2}\int_a^b F^2(\gamma(t),\dot{\gamma}(t))\, dt</math>
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| in the sense that its [[functional derivative]] vanishes among differentiable curves {{nowrap|''γ'':[''a'',''b'']→''M''}} with fixed endpoints ''γ''(''a'')=''x'' and ''γ''(''b'')=''y''.
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| ===Canonical spray structure on a Finsler manifold===
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| 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 ''TM'' as
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| :<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)
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| - \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,
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| </math>
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| where ''k''=1,...,''n'' and ''g''<sub>ij</sub> is the coordinate representation of the fundamental tensor, defined as
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| :<math>
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| g_{ij}(x,v) := g_v\left(\tfrac{\partial}{\partial x^i}\big|_x,\tfrac{\partial}{\partial x^j}\big|_x\right).
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| </math>
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| Assuming the [[Convex function#Strongly convex functions|strong convexity]] of ''F''<sup>2</sup>(''x,v'') with respect to ''v''∈''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'']→''M''}} is a geodesic of (''M'',''F'') if and only if its tangent curve {{nobreak|''γ''':[''a'',''b'']→''TM'' \0}} is an [[integral curve]] of the [[vector field|smooth vector field]] ''H'' on ''TM'' \0 locally defined by
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| :<math>
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| H|_{(x,v)} := v^i\tfrac{\partial}{\partial x^i}\big|_{(x,v)} - \ 2G^i(x,v)\tfrac{\partial}{\partial v^i}\big|_{(x,v)},
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| </math>
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| where the local spray coefficients ''G''<sup>i</sup> are given by
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| :<math>
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| G^i(x,v) := \frac{g^{ij}(x,v)}{4}\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.
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| </math>
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| The vector field ''H'' on ''TM''/0 satisfies ''JH'' = ''V'' and [''V'',''H''] = ''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 ''TM'' \0. Hence, by definition, ''H'' is a [[Spray (mathematics)|spray]] on ''M''. The spray ''H'' defines a [[Ehresmann connection|nonlinear connection]] on the [[fibre bundle]] {{nowrap|''TM'' \0 → ''M''}} through the [[Ehresmann connection|vertical projection]]
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| :<math> v:T(TM\setminus 0)\to T(TM\setminus 0) \quad ; \quad v := \tfrac{1}{2}\big( I + \mathcal L_H J \big).</math>
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| In analogy with the [[Riemannian manifold|Riemannian]] case, there is a version
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| :<math>D_{\dot\gamma}D_{\dot\gamma}X(t) + R_{\dot\gamma}(\dot\gamma(t),X(t)) = 0</math>
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| of the [[Jacobi equation]] for a general spray structure (''M'',''H'') in terms of the [[Ehresmann connection|Ehresmann curvature]] and
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| [[Double tangent bundle#Nonlinear covarient derivatives on smooth manifolds|nonlinear covariant derivative]].
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| ===Uniqueness and minimizing properties of geodesics===
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| 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''<sup>2</sup> 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 curve]]s.
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| If ''F''<sup>2</sup> is strongly convex, geodesics ''γ'' : [0, ''b''] → ''M'' are length-minimizing among nearby curves until the first point ''γ''(''s'') [[conjugate point|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 manifold|Riemannian]] case.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | editor1-last=Antonelli | editor1-first=P. L. | title=Handbook of Finsler geometry. Vol. 1, 2 | url=http://books.google.com/books?id=b2B5_IUvPJgC | publisher=Kluwer Academic Publishers | location=Boston | isbn=978-1-4020-1557-1 | mr=2067663 | year=2003}}
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| * D. Bao, [[S. S. Chern]] and Z. Shen, ''An Introduction to Riemann–Finsler Geometry,'' Springer-Verlag, 2000. ISBN 0-387-98948-X.
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| *{{Citation | last1=Cartan | first1=Elie | author1-link=Élie Cartan | title=Sur les espaces de Finsler | zbl=0006.22501 | year=1933 | journal=C. R. Acad. Sci., Paris | volume=196 | pages=582–586}}
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| * [http://www.ams.org/notices/199609/chern.pdf S. Chern: ''Finsler geometry is just Riemannian geometry without the quadratic restriction'', Notices AMS, 43 (1996), pp. 959–63.]
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| *{{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))
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| * H. Rund. ''The Differential Geometry of Finsler Spaces,'' Springer-Verlag, 1959. ASIN B0006AWABG.
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| * Z. Shen, ''Lectures on Finsler Geometry,'' World Scientific Publishers, 2001. ISBN 981-02-4531-9.
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| ==External links==
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| * {{springer|title=Finsler space, generalized|id=p/f040420}}
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| * Z. Shen's [http://www.math.iupui.edu/~zshen/Finsler/ Finsler Geometry Website].
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| * [http://finsler.blogspot.com/ The (New) Finsler Newsletter]
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| [[Category:Finsler geometry]]
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| [[Category:Smooth manifolds]]
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