Finsler manifold

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, and (usually) 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

::$L(w)=frac\{1\}\{2\}|w|^2$

: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.

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

:$int\; left|frac\{dgamma\}\{dt\}(t)\; ight|,\; dt.$

Length is invariant under reparametrization. Assuming the above condition on the Hessian, geodesics are locally length-minimizing curves with constant speed, or equivalently, curves whose energy function

:$int\; left|frac\{dgamma\}\{dt\}(t)\; ight|^2,\; dt$

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

ee also

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

External links

* Z. Shen's [http://www.math.iupui.edu/~zshen/Finsler/ Finsler Geometry Website] .

References

* D. Bao, S.S. Chern and Z. Shen, "An Introduction to Riemann-Finsler Geometry," Springer-Verlag, 2000. ISBN 0-387-98948-X.
* [http://www.ams.org/notices/199609/chern.pdf 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.