Finsler manifold

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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 T_xM, the Hessian of the function L:T_xM → 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.

Contents 1 Examples 2 Geodesics 3 External links 4 References

[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.