Differential geometry, topology and fibre bundles
Differential geometry, topology and fibre bundles
This chapter introduces all necessary mathematical concepts. Section 2.1 briefly summarizes some topological definitions. Section 2.2 explains the homotopy of maps and the homotopy of groups. Section 2.3 introduces the concept of differentiable manifolds while Section 2.4 presents the differential forms together with their Hodge duals, along with the differentiation and integration. Section 2.5 discusses homology and de Rham cohomology. Section 2.6 explains important concepts such as pullback of a differential form the Lie derivative, the Lie group, and the Lie algebra. Finally, Section 2.7 constructs fibre bundles including connection and curvature, which turn out to be a suitable mathematical concept to describe the physics of gauge theories.
Keywords: homotopy, differentiable manifolds, Hodge Duals, homology, de Rham cohomology, fibre bundles
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