Prolog

1. Prolog Website: http://ckw.phys.ncku.edu.tw Homework submission: 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

2. The Geometry of Physics, An Introduction, 2nd ed. T. Frankel Cambridge University Press (97, 04) • Manifolds, Tensors, & Exterior Forms • Geometry & Topology • Lie Groups, Bundles, & Chern Forms

3. I. Manifolds, Tensors, & Exterior Forms • Manifolds & Vector Fields • Tensors, & Exterior Forms • Integration of Differential Forms • The Lie Derivative • The Poincare Lemma & Potentials • Holonomic & Nonholonomic Constraints

4. II. Geometry & Topology • R3 and Minkowski Space • The Geometry of Surfaces in R3 • Covariant Differentiation & Curvature • Geodesics • Relativity, Tensors, & Curvature • Curvature & Simple Connectivity • Betti Numbers & De Rham's Theorem • Harmonic Forms

5. III. Lie Groups, Bundles, & Chern Forms • Lie Groups • Vector Bundles in Geometry & Physics • Fibre Bundles, Gauss-Bonnet, & Topological Quantization • Connections & Associated Bundles • The Dirac Equation • Yang-Mills Fields • Betti Numbers & Covering Spaces • Chern Forms & Homotopy Groups

6. 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)

7. 1. Manifolds & Vector Fields 1.1. Submanifolds of Euclidean Space 1.2. Manifolds 1.3. Tangent Vectors & Mappings 1.4. Vector Fields & Flows

8. 1.1. Submanifolds of Euclidean Space 1.1.a. Submanifolds of RN. 1.1.b. The Geometry of Jacobian Matrices: The "differential". 1.1.c. The Main Theorem on Submanifolds of RN. 1.1.d. A Non-trivial Example: The Configuration Space of a Rigid Body

9. 1.1.a. Submanifolds of RN • A subset M = Mn Rn+ris an n-D submanifold of Rn+rif • 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