Differential Geometry in Physics

Abstract

Differential geometry is a mathematical field which lies at the foundation of many theories in physics, such as general relativity, cosmology and string theory. A good understanding of these physical theories therefore requires one to be familiar with numerous techniques and mathematical constructs within this field, as well as their implementation. In this project we study this implementation in detail, by first developing the relevant mathematical tools and then applying them to the three fields named above. The main results include the derivation of Einstein's equation in vacuum and its extension to non-empty universes, the FLRW cosmological model and the Bianchi classification of anisotropic universes. We also discuss the ADE classification of semi-simple Lie algebras and their appearance in string theory, which is manifested by the ADE-ALE correspondence.