Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is a much more general structure than a Riemannian metric. A Finsler structure on a manifold M is a function F : TM → [0,∞) such that:
1. F(x, my) = |m|F(x,y) for all x, y in TM,
2. F is infinitely differentiable in TM − {0},
3. The vertical Hessian of F2 is positive definite.
Taken From http://en.wikipedia.org/wiki/Differential_geometry
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