微分幾何二   (90年度 下學期)

1. Tensor theory: covariant derivative, curvature tensor, Stoke's formula, cf. Chap. 3,4 (I). 2. Hamiltonian system and symplectic structure: Euler-Lagrange equations, Jacobi fields, Hamiltonian system, symplectic formalism, cf. Chap. 5 (I); §28 Chap. 7 (II). 3. Topology of differentiable manifolds: fundamental groups, degree of maps, homotopy exact sequence, cf. Chap. 3,4,5 (II). 4. Geometry of differentiable manifolds: Rauch comparison theorem, Cartan-Hadamard theorem, Cartan-Ambrose-Hicks theorem, cf. Chap. 1〔CE〕. 5. De Rham cohomology, curvature formulas for Chern classes, Poincare duality, an introduction to simplicial homology, cf. Chap. 1 (III). 6. Hodge theory in Riemannian case, cf. Chap. 6〔W〕Chap.0〔G〕. 7. Critical point theory and Morse inequalities, cf. Chap. 2 (III). . Complex manifolds: Kahler geometry, line bundles, divisors, cf. Chap.0, 1〔G〕. . Lie groups and Lie algebras: symmetric spaces, SU (2)-representation theory, cf. 〔H〕.

〔I,II,III〕B. A. Dubrovin, A. T. Formenko, S. P. Novikov, Mordern Geometry. 〔CE〕 J. Cheeger, D. Ebin, Comparison theorems in Riemannian Geometry. 〔G〕 P. Griffiths, J. Harris, Principles of algebraic geometry. 〔W〕 F. Warner, Foundation of differentiable manifolds and Lie groups. 〔H〕 S. Helgason, Differential geometry, Lie groups and symmetric spaces.

上課有習題，每學期有一次測驗。 四.其他