An introduction to the theory of generalized conics and their applications

A generalized conic is a set of points with the same average distance from the pointset ΓΓ in the Euclidean coordinate space. The measuring of the average distance is realized via integration over ΓΓ as the set of foci. Using generalized conics we give a process for constructing convex bodies which are invariant under a fixed subgroup GG of the orthogonal group in RnRn. The motivation is to present the existence of non-Euclidean Minkowski functionals with G⊂O(n)G⊂O(n) in the linear isometry group provided that the closure of GG is not transitive on the unit sphere. As an application, consider RnRn as the tangent space at a point of a connected Riemannian manifold MM and GG as the holonomy group. If the holonomy group is not transitive on the unit sphere in the tangent space, then the Lévi-Civita connection is (re)metrizable in the sense that there is a smooth collection of non-Euclidean Minkowski functionals on the tangent spaces such that it is invariant under parallel transport with respect to the Lévi-Civita connection (according to Berger’s list of possible Riemannian holonomy groups, all of them are transitive on the unit sphere in the tangent space except in the case where the manifold is a symmetric space of rank≥2). We present the (re)metrizability theorem in a more general context of metrical linear connections with a torsion tensor that is not necessarily vanishing. This allows us to declare eight classes of manifolds equipped with an invariant smooth collection of Minkowski functionals on the tangent spaces. They are called Berwald manifolds in a general sense.