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Finsler manifold

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. named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation . A Finsler manifold is a differentiable manifold M together with a Finsler metric, which is a continuous nonnegative function F: TM → [0, +∞) defined on the tangent bundle so that for each point x of M, F(v + w) ≤ F(v) + F(w) for every two vectors v,w tangent to M at x (subadditivity). F(λv) = λF(v) for all λ ≥ 0 (but not necessarily for λ < 0) (positive homogeneity). F(v) > 0 unless v = 0 (positive definiteness). In other words, F(x, −) is an asymmetric norm on each tangent space TxM. The Finsler metric F is also required to be smooth, more precisely: F is smooth on the complement of the zero section of TM. The subadditivity axiom may then be replaced by the following strong convexity condition: For each tangent vector v ≠ 0, the Hessian matrix of F2 at v is positive definite. Here the Hessian of F2 at v is the symmetric bilinear form also known as the fundamental tensor of F at v. Strong convexity of F implies the subadditivity with a strict inequality if ≠ . If F is strongly convex, then it 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. 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.

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