I've attempted to get a bead on Finsler manifolds before attending an upcoming seminar that involves them. I've done some reading in the literature and think I understand that they are similar to Riemannian manifolds, but that the metric tensor need not be a quadratic differential and need not be isotropic. Also, the "sphere" of unit tangent vectors at a point in a Finsler manifold need not be symmetric although it does have to be convex.
I'd love to hear a nice, illustrative, concrete example of a Finsler manifold, especially if it could involve a picture and an explanation of why it's interesting to you.
Thanks!