Lectures: Wednesday, 10:20 ~ 12:10, and Friday 10:20 ~ 11:10; at Astro-Math 102
Office hour: Monday, 14:00 ~ 15:00 at Astro-Math 458
Course Assistant: 連焌凱
Problem session: Friday 11:20 ~ 12:10; at Astro-Math 102
Course prerequisite: linear algebra and mathematical analysis
It will be helpful if you are familiar with general topology. (more precisely, ch.1 of
Topology and Geometry by Glen Bredon
mathscinet)
Grading scheme:
Homework 30%
You have two jokers: the lowest two grades will be discarded.
Midterm 35% on October 22
Final Exam 35% on December 17
Main Topics:
Theory of surfaces
Differentiable manifolds
Differential forms
Textbooks:
Felix Schulze, Geometry of Curves and Surfaces. file
John M. Lee, Introduction to Smooth Manifolds. file (via NTU IP)
Other references:
Manfredo do Carmo, Differential Geometry of Curves and Surfaces.
Theodore Shifrin, Differential Geometry: A First Course in Curves and Surfaces. file
Loring W. Tu, An Introduction to Manifolds. file (via NTU IP)
Lecture summaries and references
(Week 1) parametrized curves; rectifiable and arclength; Frenet frame for curves in R^3; Hopf Umlaufsatz; isoperimetric inequality. ref: [Schulze] p.5 ~ 23
(Week 2) regular surface; applications of IFT; change of parametrization; smooth function and its differential; tangent plane; 1st fundamental form. ref: [Schulze] p.26 ~ 39
(Week 3) smooth map between regular surface, and its differential; Weingarten map and 2nd fundamental form; mean curvature and Gauss curvature; compatibility: Gauss equation and Codazzi-Mainardi equation. ref: [Schulze] p.32 ~ 34, 39 ~ 42, 49 ~ 52
(Week 4) Gauss Theorema Egregium; principal curvatures as the coefficients in Taylor series; totally umbilical surface; vector fields and covariant derivative; change of parametrization and covariant derivative; covariant derivative along a curve. ref: [Schulze] p.53 ~ 62
(Week 5) parallel transport; geodesic and variation of energy; geodesic exponential map; geodesic curvature and normal curvature. ref: [Schulze] p.60 ~ 70
(Week 6) Gauss-Bonnet theorem: local and global. ref: [Schulze] p.70 ~ 77
(Week 7) mean curvature and variation of the area; hyperbolic geometry. ref: [Shifrin] p.91 ~ 110
smooth manifold. ref: [Lee] ch.1
(Week 8) Midterm
(Week 9) derivation and tangent plane; smooth map between manifolds, and its differential; tangent bundle; immersion, submersion, embedding, submanifold. ref: [Lee] ch.2, 3, 4
(Week 10) basic property of immersion and submersion; embedding into Euclidean spaces, and (weak) Whitney embedding theorem. ref: [Lee] ch.6
(Week 11) vector fields; integral curves and flows; commuting derivatives and Lie brackets; Lie derivative; bracket and commutation of the flow. ref: [Lee] ch.8, 9, 19
(Week 12) involutive distribution and Frobenius theorem; multi-linear algebra: tensor and alternating algebra; wedge product; differential forms. ref: [Lee] ch.19, 14
(Week 14) integration on manifolds; Stokes theorem; de Rham cohomology and homomorphism to R^1; homotopy invariance of de Rham cohomology. ref: [Lee] ch.16, 17
(Week 15) Mayer-Vietoris sequence for de Rham cohomology; Frobenius theorem in terms of differential ideal. ref: [Lee] ch.17, 19