Mathematical Foundations of Theoretical Physics

These notes present the geometric foundations of modern theoretical physics in a mathematically rigorous way. Starting from topology, they currently cover smooth manifolds and tangent vectors. Future sections will extend the discussion to fibre bundles, with the aim of providing the mathematical background for gauge theories.

These notes also served as the basis for a corresponding lecture at Goethe University Frankfurt in 2025. 

ADM Formalism

These notes present a rigorous, geometric formulation of the ADM (3+1) decomposition of General Relativity. Starting from a spacetime foliation by Cauchy hypersurfaces, the construction develops the induced geometry, projection operators, intrinsic covariant derivative, and the Gauss–Codazzi–Ricci equations in a systematic and self-contained way. This framework is then used to derive the Hamiltonian formulation of gravity, including the constraint equations and their algebra. 

Information Geometry

These notes introduce information geometry starting from a divergence function as the primary object. Under suitable regularity assumptions, the metric, dual affine connections, and induced geometric structures of a statistical manifold are obtained directly from the divergence. This viewpoint unifies and generalizes standard constructions such as the Fisher information metric and alpha-connections.