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

The first main work of this paper is to generalize intrinsic geometry, that is: (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 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 on Hilbert's 6th problem at the most basic level. (1) It becomes true that principles, postulates and artificially introduced equations of fundamental physics are turned into theorems which automatically hold in intrinsic geometrical theory. (2) Gravitatinoal field and gauge field are 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) The mathematical foundations of general relativity and quantum mechanics achieve unification. Intrinsic geometry makes them have the same view of time and space and unified description of evolution