Prolog Website Homework submission http ckw phys ncku
Prolog Website: Homework submission: • • http: //ckw. phys. ncku. edu. tw class@ckw. phys. ncku. edu. tw Line, surface & volume integrals in n-D space → Exterior forms Time evolution of such integrals → Lie derivatives Dynamics with constraints → Frobenius theorem on differential forms Curvatures → Differential geometry – Spacetime curvatures ~ General relativity – Field space curvatures ~ Gauge theories • Symmetries of quantum fields → Lie groups • Existence & uniqueness of problem → Topology – Examples: Homology groups, Brouwer degree, Hurewicz homotopy groups, Morse theory, Atiyah-Singer index theorem, Gauss-Bonnet. Poincare theorem, Chern characteristic classes
The Geometry of Physics, An Introduction, 2 nd ed. T. Frankel Cambridge University Press (97, 04) I. Manifolds, Tensors, & Exterior Forms II. Geometry & Topology III. Lie Groups, Bundles, & Chern Forms
I. Manifolds, Tensors, & Exterior Forms 1. 2. 3. 4. 5. 6. Manifolds & Vector Fields Tensors, & Exterior Forms Integration of Differential Forms The Lie Derivative The Poincare Lemma & Potentials Holonomic & Nonholonomic Constraints
II. Geometry & Topology 7. R 3 and Minkowski Space 8. The Geometry of Surfaces in R 3 9. Covariant Differentiation & Curvature 10. Geodesics 11. Relativity, Tensors, & Curvature 12. Curvature & Simple Connectivity 13. Betti Numbers & De Rham's Theorem 14. Harmonic Forms
III. Lie Groups, Bundles, & Chern Forms 15. Lie Groups 16. Vector Bundles in Geometry & Physics 17. Fibre Bundles, Gauss-Bonnet, & Topological Quantization 18. Connections & Associated Bundles 19. The Dirac Equation 20. Yang-Mills Fields 21. Betti Numbers & Covering Spaces 22. Chern Forms & Homotopy Groups
Supplementary Readings • Companion textbook: – C. Nash, S. Sen, "Topology & Geometry for Physicists", Acad Press (83) • Differential geometry (standard references) : – M. A. Spivak, "A Comprehensive Introduction to Differential Geometry" ( 5 vols), Publish or Perish Press (79) – S. Kobayashi, K. Nomizu, "Foundations of Differential Geometry" (2 vols), Wiley (63) • Particle physics: – A. Sudbery, "Quantum Mechanics & the Particles of Nature", Cambridge (86)
1. Manifolds & Vector Fields 1. 1. 1. 2. 1. 3. 1. 4. Submanifolds of Euclidean Space Manifolds Tangent Vectors & Mappings Vector Fields & Flows
1. 1. Submanifolds of Euclidean Space 1. 1. a. 1. 1. b. 1. 1. c. 1. 1. d. Submanifolds of RN. The Geometry of Jacobian Matrices: The "differential". The Main Theorem on Submanifolds of RN. A Non-trivial Example: The Configuration Space of a Rigid Body
1. 1. a. Submanifolds of RN A subset M = Mn Rn+r is an n-D submanifold of Rn+r if " P M, a neiborhood U in which P can be described by some coordinate system of Rn+r where are differentiable functions are called local (curvilinear) coordinates in U
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