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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_''x''''M'', the Hessian of the function ''L'':T_''x''''M'' → '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

Examples

Geodesics

See also

External links

References

Examples

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

★ Randers manifolds

Geodesics

The length of γ, a differentiable curve in ''M'', is given by
:
ight|, dt.
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
:
ight|^2, dt.
is extremal under functional derivatives.

See also

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

External links

★ Z. Shen's Finsler Geometry Website.

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.