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Differential Geometry I, Fall 2022

Course Information

NTU COOL
Lectures: Wednesday, 16:30 ~ 17:20 and Friday 10:20 ~ 12:10 at Astro-Math 101
Office hours: 15:00 ~ 16:00 every Wednesday
Grading scheme:
- Homework 30%
  You have two jokers: the lowest two grades will be discarded.
- Midterm 35%
- Final Exam 35%
Grader: 李宸寬
Textbook:
- [W] Frank Warner, Foundations of differentiable manifolds and Lie groups. MR
Other references:
- [dC] Manfredo do Carmo, Riemannian Geometry. MR
- [T] Clifford Taubes, Differential geometry. Bundles, connections, metrics and curvature. MR
- [W'] Chin-Lung Wang, Differential Geometry. in link

(week 1) smooth manifold. Reference: [W] §1.2, §1.3.
(week 2) derivation, tangent bundle, differential, immersion, submersion, embedding. Reference: [W] §1.4, §1.5.
(week 3) weak Whitney embedding, tangent vector field, flow, Lie bracket. Reference: [W] §1. and [W'] §1.5, §1.6.
(week 4) Lie derivative, Frobenius theorem. Reference: [W] §1.7, §1.8.
(week 5) tensor, differential form, exterior derivative. Reference: [W] §2.1, §2.2, §2.3.
(week 6) Lie derivative of differential form, manifold with boundary, orientation, integration, Stokes theorem. Reference: [W] §2.3, §4.1, §4.2.
(week 7) Frobenius theorem in terms of differential ideals, Poincaré lemma, Mayer--Vietoris sequence. Reference: [W] §2.4, §4.3.
(week 8) Midterm
(week 9) Riemannian metric, Levi-Civita connection, Riemann curvature tensor. Reference: [W'] §3.1, §3.4.
(week 10) first variational formula, geodesic, exponential map, Gauss map. Reference: [W'] §3.1, §3.2.
(week 11) Hopf-Rinow theorem, Jacobi field, geodesic normal coordinate. Reference: [W'] §3.2, §3.3.
(week 12) the method of moving frame, Hodge theory. Reference: [W] §6.1, §6.2.
(week 13) Hodge theory (continued). Reference: [W] §6.1, §6.2.
(week 14) Bochner formula, Sobolev spaces and its properties. Reference: [W] §6.3.
(week 15) elliptic operator, elliptic estimates. Reference: [W] §6.3, §6.4, §6.6.