In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

For each point x of M, and for every vector v in the tangent space T_xM, the second derivative of the function L:T_xM → R given by

at v is positive definite.

Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

The length of γ, a differentiable curve in M, is given by In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

Note that this is parametrization-invariant. Further, in Finsler geometry geodesics are curves in M whose length is extremal under functional derivatives. That is, since there is a (not necessarily quadratic) functional dependence on the line element of a Finsler manifold, then the locally minimal curves of the space (the geodesics) must also satisfy the additional variation. Thus geodesics in Finsler manifolds, though still minimal in an important way, are somehow less flat than Riemannian geodesics. However, the definition gets subsumed as all Riemannian manifolds are Finsler. In mathematics and theoretical physics, the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. ... The line element in mathematics can most generally be thought of as the square of the change in a position vector in an affine space equated to the square of the change of the arc length. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

In Riemannian geometry, geometric objects are taken to be only a function of their position, however geometric objects can be a function of many things. In particular, when geometrical objects are a function of both position and velocity, the geometry is called Finsler geometry. The question arises as to whether spacetime is described by Finsler geometry, so far there is no observational evidence that this is the case. Finsler geometries arise as the phase space of Hamiltonian mechanics. Typically the curvature will depend on both the position and momentum of a particle. If one associates the momentum with a velocity then the phase space is a Finsler space. In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... This article may be too technical for most readers to understand. ... Phase space is a useful construct in mathematics and physics to demonstrate and visualise the changes in the dynamical variables of a system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

References

• Hanno Rund. The Differential Geometry of Finsler Spaces. Springer-Verlag 1959. ASIN B0006AWABG.