Diffusion, Laplacian and Hodge Decomposition on Finsler Spaces

Abstract

Throughout this paper M will be a compact Finsler space with positive definite metric. That is to say, M will be a boundaryless compact manifold equipped with a positive smooth ℝ-valued function F defined on the slit tangent bundle TM (the tangent bundle TM with the zero section removed) such that F(x, y) is a positive homogeneous function of degree one in y ∈ T _x M for any x∈ M and the Finsler metric tensor defined by is positive definite. The latter is a tensor in the Finslerian sense, that is, such that the diagram commutes. (Here π₁ and π₂ are the projections of the bundles TM and T ² M.)