Average methods and their applications in Differential Geometry I

Abstract

In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball with its boundary formed by the unit vectors. Using one-homogeneous extension we have a so-called Minkowski functional to measure the lenght of vectors. The half of its square is called the energy function. Under some regularity conditions we can introduce an average Euclidean inner product by integrating the Hessian matrix of the energy function on the Minkowskian unit sphere. Changing the origin in the interior of the body we have a collection of Minkowskian unit balls together with Minkowski functionals depending on the base points. It is a kind of special Finsler manifolds called a Funk space. Using the previous method we can associate a Riemannian metric as the collections of the Euclidean inner products belonging to different base points. We investigate this procedure in case of Finsler manifolds in general. Central objects of the associated Riemannian structure will be expressed in terms of the canonical data of the Finsler space. We take one more step forward by introducing associated Randers spaces by averaging of the vertical derivatives of the Finslerian fundamental function. Such kind of perturbation will have a crucial role when we apply the general results to Funk spaces together with some contributions to Brickell's conjecture on Finsler manifolds with vanishing curvature tensor $\mathbb{Q}$.