prolog

Prolog

• 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

Website: http://ckw.phys.ncku.edu.tw

Homework submission: class@ckw.phys.ncku.edu.tw

The Geometry of Physics, An Introduction, 2nd 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 Forms1. Manifolds & Vector Fields

2. Tensors, & Exterior Forms

3. Integration of Differential Forms

4. The Lie Derivative

5. The Poincare Lemma & Potentials

6. Holonomic & Nonholonomic Constraints

II. Geometry & Topology

7. R3 and Minkowski Space

8. The Geometry of Surfaces in R3

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. Submanifolds of Euclidean Space

1.2. Manifolds

1.3. Tangent Vectors & Mappings

1.4. Vector Fields & Flows

1.1. Submanifolds of Euclidean Space

1.1.a. Submanifolds of N.

1.1.b. The Geometry of Jacobian Matrices:

The "differential".

1.1.c. The Main Theorem on Submanifolds of N.

1.1.d. A Non-trivial Example:

The Configuration Space of a Rigid Body

1.1.a. Submanifolds of N

A subset M = Mn n+r is an n-D submanifold of n+r if

PM, a neiborhood U in which P can be described by some coordinate system of n+r

1 1, , , ; , ,n rx y x x y y

where 1, ,j j n jy f x x f x are differentiable functions

1, , nx x x are called local (curvilinear) coordinates in U