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, smoothly depending on position.

Many works also assume the tangent space to be a Minkowsky geometry when endowed with the Banach norm, which can be stated in the following way, if one denotes by the norm of the vector w then:

For each point x of M, and for every nonzero vector v in the tangent space T_xM, the Hessian of the function given by

is positive definite at v.

Contents 1 Examples 2 Geodesics 3 See also 4 External links 5 References

[edit] Examples

• Normed vector spaces of finite dimension, such as Euclidean spaces.
• Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.
• Randers manifolds
• Hilbert Geometries

[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 whose energy function

is extremal (in the sense that its functional derivative vanishes).

[edit] See also

• Metric tensor, used for differentiable manifolds with inner-product norms.

[edit] External links

• Z. Shen's Finsler Geometry Website.
• The (New) Finsler Newsletter

[edit] References

• D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, 2000. ISBN 0-387-98948-X.
• S. Chern: Finsler geometry is just the Riemannian geometry without the quadratic restriction, Notices AMS, 43 (1996), pp. 959-63.
• H. Rund. The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959. ASIN B0006AWABG.
• Z. Shen, Lectures on Finsler Geometry, World Scientific Publishers, 2001. ISBN 981-02-4531-9.