---

Differential Geometry I, Fall 2026

---

Course Information

• NTU COOL
• Lectures: Wednesday, 13:20 ~ 14:10 at Astro-Math 102, and Friday 10:20 ~ 12:10 at Astro-Math 101
• Office hours: 14:00 ~ 15:00 every Monday
• Grading scheme:
  • Homework 20%
    You have two jokers: the lowest two grades will be discarded.
  • Midterm 40%
  • Final Exam 40%
• Grader: 連焌凱
• Textbook:
  • [L1] John M. Lee, Introduction to Smooth Manifolds, second edition. MR
  • [L2] John M. Lee, Introduction to Riemannian Manifolds, second edition. MR
  • [R] John Roe, Elliptic Operators, Topology and Asymptotic Methods, second edition. MR

Lecture summaries and references

• (week 1) smooth manifolds, smooth maps, submanifolds, examples. Reference: [L1] §1–2.
• (week 2) tangent spaces, tangent bundle, vector fields, partitions of unity. Reference: [L1] §3.
• (week 3) flows, Lie brackets, vector bundles, cotangent bundle. Reference: [L1] §10–11.
• (week 4) differential forms, exterior derivative, pullback, Lie derivative, Cartan formula. Reference: [L1] §12–14, §18.
• (week 5) orientation, integration, Stokes theorem, Poincaré lemma, de Rham cohomology. Reference: [L1] §15–17.
• (week 6) Lie groups, Lie algebras, exponential map, group actions. Reference: [L1] §7–8.
• (week 7) Sard theorem, Whitney embedding theorem, generic projections. Reference: [L1] §6.
• (week 8) distributions, integral manifolds, Frobenius theorem, foliations. Reference: [L1] §19.
• (week 9) Riemannian metrics, Levi-Civita connection, covariant derivative, geodesics. Reference: [L2] §2–5.
• (week 10) exponential map, curvature tensor, Ricci and sectional curvature, normal coordinates. Reference: [L2] §5–8.
• (week 11) Hodge star, codifferential, Hodge Laplacian, harmonic forms, Green's formula. Reference: [R] §1–2.
• (week 12) Sobolev spaces on manifolds, Rellich compactness, Sobolev embedding. Reference: [R] §5.
• (week 13) elliptic operators, elliptic regularity, finite-dimensional kernel, closed range, Green operator. Reference: [R] §5.
• (week 14) Fredholm alternative, orthogonal decomposition, proof of the Hodge decomposition. Reference: [R] §5–6.
• (week 15) Hodge theorem, harmonic representatives, Poincaré duality, Betti numbers, examples. Reference: [R] §6.

Homework

• Homework 1: due September ??.