Finsler manifold
From Wikipedia, the free encyclopedia
In mathematics, particularly differential geometry, 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 condition:
- For each point x of M, and for every nonzero vector v in the tangent space TxM, the Hessian of the function L:TxM → R given by
- is positive definite at v.
The above condition implies that the norm function satisfies the triangle inequality. The proof of this is not completely trivial.
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[edit] Examples
- Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.
- Randers manifolds
[edit] Geodesics
The length of γ, a differentiable curve in M, is given by
Length is invariant under reparametrization. Assuming the above condition on the Hessian, geodesics are locally length-minimizing curves with constant speed, or equivalently curves in whose energy function
is extremal under functional derivatives.
[edit] External links
Z. Shen's Finsler Geometry Website.
[edit] References
- H. Rund. The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959. ASIN B0006AWABG.
- D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, 2000. ISBN 0-387-98948-X.
- Z. Shen, Lectures on Finsler Geometry, World Scientific Publishers, 2001. ISBN 981-02-4531-9.
