A generalization of intrinsic geometry and its application to Hilbert's 6th problem

Abstract

The first main work of this paper is to generalize intrinsic geometry. (1) Riemannian manifold is generalized to geometrical manifold. (2) The expression of Erlangen program is improved, and the concept of intrinsic geometry is generalized, so that Riemannian intrinsic geometry which is based on the first fundamental form becomes a subgeometry of the generalized intrinsic geometry. The Riemannian geometry is thereby incorporated into the geometrical framework of improved Erlangen program. (3) The important concept of simple connection is discovered, which reflects more intrinsic properties of manifold than Levi-Civita connection. The second main work of this paper is to apply the generalized intrinsic geometry to Hilbert's 6th problem at the most basic level. (1) It starts from an axiom and makes key principles, postulates and artificially introduced equations of fundamental physics all turned into theorems which automatically hold in intrinsic geometrical theory. (2) Intrinsic geometry makes gravitational field and gauge field unified essentially. Intrinsic geometry of external space describes gravitational field, and intrinsic geometry of internal space describes typical gauge field. They are unified into intrinsic geometry. (3) Intrinsic geometry makes gravitational theory and quantum mechanics have the same view of time and space and unified description of evolution.