Manifolds, Groups, Bundles, and SpacetimeManifolds, Groups, Bundles, and Spacetime was written for those who are interested in modern differential geometry and its applications in physics. The primary material is suitable for a graduate level course in the theory of differentiable manifolds, Lie groups, and fiber bundles. The first two chapters are an introduction to concepts from linear algebra and tensors and can be read to establish familiarity with the notation and conventions of the text by those who are already familiar with these topics. The third and fourth chapters are a review of topics from advanced calculus and topology and are included primarily as a convenient reference. |
Contents
TOPICS FROM LINEAR ALGEBRA | 1 |
MULTILINEAR ALGEBRA | 25 |
DIFFERENTIAL CALCULUS ON VECTOR SPACES | 57 |
TOPOLOGY | 103 |
DIFFERENTIABLE MANIFOLDS | 139 |
DIFFERENTIAL CALCULUS ON MANIFOLDS | 165 |
LIE GROUPS AND LIE ALGEBRAS | 207 |
1 | 229 |
1 | 277 |
Spacetime | 291 |
3 | 304 |
THE DIFFERENTIAL GEOMETRY OF SPACETIME | 310 |
INTEGRATION ON MANIFOLDS | 317 |
A PICARDS THEOREM | 379 |
| 399 | |
FIBER BUNDLES | 237 |
Common terms and phrases
arbitrary atlas called change of basis compact components connection 1-form converges coordinate functions coordinate system Corollary countable covariant defined Definition denoted derivative Df(x diffeomorphism differentiable manifold domain dual basis dx² dx³ equation Example Exercise Əxi f is continuous geodesic gives Gl(n Hence homomorphism inner product space integral curve Lemma Let f Let G Let X1 Lie algebra Lie group linear mapping linearly independent manifold and let maximal integral curve metric tensor n-dimensional manifold non-empty normed vector space notation open set open subset orientation orthonormal p-form principal fiber bundle proof of Theorem Prove Remark respectively right-hand side S₁ scalars sequence Show space and let spacetime subspace summation surjective tensor field topological space topology transformation unique vector bundle მ მ მე