# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Finsler geometry

(Redirected from Finsler manifold)

In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property:

For each point x of M, and for every vector v in the tangent space TxM, the second derivative of the function L:TxM->R given by
$L(\bold{w})=\frac{1}{2}\|w\|^2$
at v is positive definite.

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

The length of γ, a differentiable curve in M is given by

$\int \left\|\frac{d\gamma}{dt}(t)\right\| dt$.

Note that this is reparametrization-invariant. Geodesics are curves in M whose length is extremal under functional derivatives.

03-10-2013 05:06:04