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In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional : where ''F''(''x'', · ) is a Minkowski norm (or at least an asymmetric norm) on each tangent space ''T''''x''''M''. Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an inner product (metric tensor). 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 function ''F'' defined on the tangent bundle of ''M'' so that for all tangent vectors ''v'', * ''F'' is smooth on the complement of the zero section of ''TM''. * ''F''(''v'') ≥ 0 with equality if and only if ''v'' = 0 (positive definiteness). * ''F''(λ''v'') = λ''F''(''v'') for all λ ≥ 0 (but not necessarily for λ < 0) (homogeneity). * ''F''(''v'' + ''w'') ≤ ''F''(''v'') + ''F''(''w'') for all ''w'' at the same tangent space with ''v'' (subadditivity). In other words, ''F'' is an asymmetric norm on each tangent space. Typically one replaces the subadditivity with the following strong convexity condition: * For each tangent vector ''v'', the hessian of ''F''2 at ''v'' is positive definite. Here the hessian of ''F''2 at ''v'' is the symmetric bilinear form : also known as the fundamental tensor of ''F'' at ''v''. Strong convexity of ''F''2 implies the subadditivity with a strict inequality if ''u''/''F''(''u'') ≠ ''v''/''F''(''v''). If ''F''2 is strongly convex, then ''F'' is a Minkowski norm on each tangent space. A Finsler metric is reversible if, in addition, * ''F''(−''v'') = ''F''(''v'') for all tangent vectors ''v''. A reversible Finsler metric defines a norm (in the usual sense) on each tangent space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Finsler manifold」の詳細全文を読む スポンサード リンク
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