Topology and Geometry for PhysicsA concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation. |
Contents
1 Introduction | 1 |
2 Topology | 10 |
3 Manifolds | 55 |
4 Tensor Fields | 96 |
5 Integration Homology and Cohomology | 115 |
6 Lie Groups | 173 |
7 Bundles and Connections | 205 |
8 Parallelism Holonomy Homotopy and Cohomology | 247 |
9 Riemannian Geometry | 299 |
Compendium | 347 |
| 379 | |
| 381 | |
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Common terms and phrases
base boundary called canonical closed cohomology compact complex connected components connection form considered continuous coordinate neighborhood coordinate system corresponding cotangent vector countable covariant curvature curve denoted derivative diffeomorphism differential dimension elements equations equivalent exterior Fermi finite function F gauge field geodesic geometry Gl(m Gl(n hence homology homomorphism homotopy classes horizontal integral invariant isomorphic Lie algebra Lie group Lie group G Lie subgroup linear connection loop manifold matrix metric n-dimensional non-zero norm normed vector space open sets path pathwise connected physics polyhedron principal fiber bundle properties r-form real function relation relative topology Riemannian Sect sequence simply connected smooth structure submanifold subset tangent space tangent vector tangent vector field tensor field theorem theory topological space transformation trivial uniquely defined vector field vector space yields zero