Finsler manifold

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In mathematics, a Finsler manifold is a differentiable manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth, usually it is assumed to satisfy the following regularity condition:

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

:[L(w)=\frac\|w\|^2]

at v is positive definite.

Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

The length of γ, a differentiable curve in M, is given by

[\int \left\|\frac(t)\right\| dt.]

Length is invariant under reparametrization. With the above regularity condition, geodesics are locally length-minimizing curves with constant speed, or equivalently curves in whose energy function

[\int \left\|\frac(t)\right\|^2 dt.]

is extremal under functional derivatives.

References

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

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