Preliminaries on Differentiable Manifolds and Lie groups. Homotopy and Homology. Principale bundles, Morphisms, Reductions, Induced Bundles. Associated Fiber Bundles, Vector Bundles, Gauge Group. Connections on Principal Bundles, Parallel Transport, Vector Bundle Valued Forms, Curvature Form, Bianchi's Identity, Holonomy and the Ambrose-Singer Theorem, Linear and Riemannian Connections, Weil Homomorphim, Characteristic Classes, Constructions a la Kaluza-Klein.
M. Nakahara, Geometry, Topology and Physics, Second Edition, Graduate Student Series in Physics, CRC Press; 2 edition (June 6, 2003)
22. De Bartolomeis, Paolo Principal bundles in action. Conference on Differential Geometry and Topology (Italian) (Parma, 1991). Riv. Mat. Univ. Parma (4) 17* (1991), 1–65 (1993)
Learning Objectives
We aim at giving some advanced notions in differential topology and geometry; these concepts might be then useful in approaching and developing further more specific arguments in theoretical and mathematical physics.
Prerequisites
Linear algebra, multivariables calculus, elements of point set topology.
Teaching Methods
Lectures using a blackboard.
Type of Assessment
Oral examination.
Course program
0. Main Topic: Connection Theory on Principal Bundles.
1. Preliminaries on the Theory of Differentiable Manifolds and Lie Groups.
2. Homotopy and Homology.
3. Principal Bundles: definitions and fundamental examples, Transitions Functions, Local Sections and Local Trivialization, Morphisms, Reductions, Induced Fiber Bundles.
4. Associated Bundles: definitions and fundamental examples, Vector Bundles, Gauge Group.
5. Connections on Principal bundles: definition and fundamental examples, Connection Forms, Horizontal Lifts and Parallel Transport, Pseudotensorial Forms and Tensorial Forms, Vector bundles Valued Forms, Exterior Covariant Derivative, Curvature Form and its Geometric Interpretations, Bianchi's Identity, Holonomy and Ambrose-Singer Theorem, Linear Connections, Riemannian Connections, Weil Homomorphism, Charachteristic Classes, Constructions a la Kaluza-Klein.