About: Finsler manifold

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In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM, that enables one 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.

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rdf:type	yago:WikicatSmoothManifolds yago:Artifact100021939 yago:Conduit103089014 yago:Manifold103717750 yago:Object100002684 yago:Passage103895293 yago:PhysicalEntity100001930 yago:Pipe103944672 yago:YagoGeoEntity yago:YagoPermanentlyLocatedEntity yago:Tube104493505 yago:Way104564698 yago:Whole100003553
rdfs:label	Finsler-Mannigfaltigkeit (de) Finsler manifold (en) Espace de Finsler (fr) 핀슬러 다양체 (ko) Finsler-variëteit (nl) Финслерова геометрия (ru) Variedade de Finsler (pt) 芬斯勒流形 (zh) Фінслерова геометрія (uk)
rdfs:comment	In der Geometrie sind Finsler-Mannigfaltigkeiten eine Verallgemeinerung riemannscher Mannigfaltigkeiten. Sie sind nach Paul Finsler benannt. (de) Un espace de Finsler est une variété différentielle possédant une métrique asymétrique locale, c'est-à-dire une (en) sur le fibré tangent. Les variétés de Finsler sont donc une généralisation des variétés de Riemann. (fr) 미분기하학에서 핀슬러 다양체(영어: Finsler manifold)는 리만 다양체의 일반화이다. 각 접공간 위에 양의 정부호 대칭 쌍선형 형식이 주어진 리만 다양체와 달리, 대신 (일반화) 노름이 주어진다. (ko) Фи́нслерова геометрия — одно из обобщений римановой геометрии.В финслеровой геометрии рассматриваются многообразия с финслеровой метрикой; то есть выбором нормы на каждом касательном пространстве, которая гладко меняется от точки к точке. (ru) Фінслерова геометрія — одне з узагальнень ріманової геометрії. У фінслеровій геометрії розглядаються многовиди з фінслеровим метричним тензором; тобто вибором гладкої норми на кожному дотичному просторі, яка гладко змінюється від точки до точки. (uk) 芬斯勒流形(Finsler manifold)或芬斯勒几何(Finsler geometry)由瑞士數學家(Paul Finsler)在1918年的博士论文中提出。芬斯勒几何可以说是黎曼几何在取消掉二次型限制后的版本，与变分学密切相关。芬斯勒几何分為實數芬斯勒几何与複數芬斯勒几何，且在生物、工程、物理等领域有广泛应用。芬斯勒流形是一个在每个切空间有巴拿赫(Banach)範數的微分流形，该巴拿赫范数是位置的光滑函数并满足以下条件：对M中的每点x，对切空间中的每一向量v，如下函数的二阶导数在v是正定的。黎曼流形(但不包括伪黎曼流形)是芬斯勒流形在嘉当张量处处为零时的特例。可微曲线γ的长度由下式给出注意这是一个重参数化不变量，测地线是长度在变分时取极值的曲线。 (zh) In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM, that enables one 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. (en) In de differentiaalmeetkunde, een deelgebied van de wiskunde, is een Finsler-variëteit een differentieerbare variëteit samen met de structuur van een intrinsieke , waar de lengte van enige γ:[a,b]→M wordt gegeven door de lengtefunctionaal waar F(x,·) een Minkowski-norm (of ten minste een ) op elke raakruimte TxM is. Finsler-variëteiten veralgemenen Riemann-variëteiten door niet langer aan te nemen dat deze infinitesimaal Euclidisch zijn in de zin dat de (asymmetrische) norm op elke raakruimte wordt geïnduceerd door een inwendig product (metrische tensor). (nl) Em Matemática, particularmente em geometria diferencial, uma Variedade de Finsler é uma variedade diferenciável juntamente com a estrutura de uma estrutura de um em que o comprimento de qualquer curva retificável γ : [a,b] → M é dada pelo comprimento funcional Élie Cartan chamou de variedade de Finsler porque Paul Finsler, foi quem estudou esta geometria em sua dissertação. (pt)
dcterms:subject	Smooth manifolds Differential geometry Riemannian manifolds Finsler geometry Riemannian geometry
Wikipage page ID	385510 (xsd:integer)
Wikipage revision ID	1118104389 (xsd:integer)
Link from a Wikipage to another Wikipage	Bilinear form Smooth manifolds Homogeneous function Hopf–Rinow theorem Vector field Double tangent bundle Integral curve Intrinsic metric Quasimetric Quasimetric space Differential geometry Mathematics Norm (mathematics) Notices of the American Mathematical Society Riemannian manifolds Ehresmann connection Einstein notation Geodesic Minkowski functional Convex function Smooth curve Smooth function Comptes rendus de l'Académie des Sciences Finsler geometry Functional derivative Spray (mathematics) Positive-definite function Euler–Lagrange equation Fibre bundle Parametrization (geometry) Differential structure Pseudo-Riemannian manifold Riemannian manifold Asymmetric norm Riemannian geometry Hessian matrix Symmetric tensor Differentiable curve Differentiable manifold Differential form Differential geometry Positive-definite matrix Inner product Subadditivity Inverse matrix Normed vector space Paul Finsler Conjugate point Jacobi equation
Link from a Wikipage to an external page	http://finsler.blogspot.com/ https://books.google.com/books%3Fid=b2B5_IUvPJgC https://www.ams.org/notices/199609/chern.pdf%7Cmr=1400859
sameAs	Finsler manifold Finsler manifold Finsler manifold Finsler manifold Finsler manifold Finsler manifold Finsler manifold Finsler manifold Finsler manifold Finsler manifold

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