Topics in Differential Geometry

Abstract

Abstract This article assumes a basic knowledge of differentiable manifolds, vector fields, and differential forms (such as that provided in the article DIFFERENTIABLE MANIFOLDS ). The first topic covered is the theory of integration of forms on oriented manifolds, whose most important result, generalizing and encompassing all the classical theorems of vector calculus, is the theorem of Stokes. The second topic is an introduction to Lie groups, namely, differentiable manifolds with a differentiablegroup structure. Important in themselves as the expression of symmetries in physical theories, Lie groups are also a necessary ingredient for the treatment of fiber bundles, which are manifolds with extra structure. Among the topics covered within this realm, principal bundles, linear connections, torsion, and curvature are particularly important in applications.