Manifold Theory: An Introduction for Mathematical PhysicistsThis account of basic manifold theory and global analysis, based on senior undergraduate and post-graduate courses at Glasgow University for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology. Topology is included in two appendices because many courses on mathematics for physics students do not include this subject.
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Contents
Tensor algebra | 26 |
Exercises 2 | 58 |
Vector and tensor fields on a manifold | 88 |
Exterior differential forms | 114 |
Exercises 5 | 142 |
3 | 154 |
Exercises 6 | 177 |
Symplectic manifolds | 239 |
Exercises 9 | 282 |
Exercises 10 | 301 |
Complex linear algebra Almost complex manifolds | 349 |
Anayltic topology | 375 |
Quaternions and Cayley numbers | 396 |
Homotopy review | 404 |
Some answers some hints and some fragmentary solutions to the exercises | 412 |
419 | |
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Common terms and phrases
affine connexion atlas axi axi base space basis chart compact complex manifold complex structure components continuous mapping contravariant vector coordinate domain coordinate system covariant derivative covariant tensor covector curvature tensor curve defined definition denoted diffeomorphism differentiable manifold ds² E₁ element equations Example exists Əxi follows function G₁ G₂ geodesic given GL(n Hence Hermitian homeomorphism identity inner product integral isomorphism Kählerian Let f Lie algebra Lie group linear linearly independent Lorentz manifold of dimension matrix metric space metric tensor neighbourhood non-zero numbers obtained open ball open sets orientable p-form paracompact parallel transport permutation Proof Let Prove pseudo-Riemannian manifold regular submanifold respectively Riemannian scalar Show smooth manifold subgroup subset subspace symmetric symplectic tangent vector tensor field tensor of type Theorem topological space U₁ U₂ V₁ vector field vector space zero