Classical mechanics can be reformulated through . The phase space of a physical system is treated as a symplectic manifold.

The Standard Model is essentially a study of geometry over principal bundles with specific symmetry groups ( 3. Hamiltonian Mechanics and Symplectic Geometry

The Riemann curvature tensor and Ricci tensor are used to relate the geometry of spacetime to the energy and momentum of the matter within it via the Einstein Field Equations. 2. Gauge Theory and Fiber Bundles

Modern particle physics relies on , which is geometrically described using fiber bundles . In this framework: Fields are sections of bundles.