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

<math>L(w)=\frac{1}{2}\|w\|^2</math>

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

<math>\int \left\|\frac{d\gamma}{dt}(t)\right\| dt.</math>

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

<math>\int \left\|\frac{d\gamma}{dt}(t)\right\|^2 dt.</math>

is extremal under functional derivatives.

References

• Hanno Rund. The Differential Geometry of Finsler Spaces. Springer-Verlag 1959. ASIN B0006AWABG.zh:芬斯勒几何