A continuous real function 
defined on the tangent
bundle 
of an 
-dimensional
smooth manifold 
is said to be a Finsler metric if
1. 
is differentiable
at 
,
2. 
for any element
and any real
number 
,
3. Denoting the metric
![g_(ij)(x,y)=1/2(partial^2[L(x,y)]^2)/(partialy^ipartialy^j),](/images/equations/FinslerMetric/NumberedEquation1.svg) |
then 
is a positive definite matrix.
A smooth manifold 
with a Finsler metric is called a Finsler
space.
