Differential Geometry II, Spring 2023

Course Information

• NTU COOL
• Lectures: Wednesday, 16:30 ~ 17:20 at Astro-Math 101,
  and Friday 10:20 ~ 12:10 at Astro-Math 102
• 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 and 高尉庭
• Main Topics:
  • vector bundle: connection, curvature, characteristic class
  • minimal submanifold
  • heat kernel and isometric embedding
• References:
  • [1] S. Morita, Geometry of differential forms (ch. 5)
  • [2a] notes by R. Schoen, file
  • [2b] notes by C.-L. Wang, file
  • [3a] S. Rosenberg, The Laplacian on a Riemannian manifold: An introduction to analysis on manifolds. London Mathematical Society Student Texts, 31. 1997.
  • [3b] P. Bérard & G. Besson & S. Gallot, Embedding Riemannian manifolds by their heat kernel, paper
  • [3c] P. Bérard, Spectral geometry: direct and inverse problems, Springer Lecture Notes in Math. 1207, 1986.

Lecture summaries and references

• (week 1) vector bundle: definition and basic examples, some constructions. Reference: [1] §5.1.
• (week 2) vector bundle: more constructions, connections. Reference: [1] §5.2.
• (week 3) vector bundle: curvature, parallel transport and horizontal distribution. elementary Lie groups. Reference: [1] §5.3 and §5.4.
• (week 4) characteristic class: Chern and Pontryagin classes. Reference: [1] §5.4 and §5.5.
• (week 5) Euler class. Reference: [1] §5.6 and §5.7.
  geometry of submanifold: fundamental equations. Reference: ch.6 of do Carmo.
• (week 6) coordinate expression of the Gauss, Codazzi and Ricci equations, first variational formula, minimal submanifolds in the Euclidean spaces, monotonicity formula, graphical hypersurface and calibration argument. Reference: [2a] §1, §2 and §3.
• (week 7) conformal structure in 2Dn Weierstrass representationn Bernstein theorem Reference: [2b] §6.2 and §6.3.
• (week 8) second variational formula. Reference: [2a] §4.
• (week 9) Jacobi operator, stability in the hypersurface case, (2d) Bernstein theorem through stability, a glimpse at higher dimensional situation. Reference: [2a] §3, §4 and §5.
• (week 10) solution to the classical Plateau problem by Douglas. Reference: [2a] §11 and §12. [2b] §6.4.
• (week 11) coarse estimate on the eigenvalues, heat kernel on Riemannian manifolds. Reference: [3a] §3.1 and §3.2. note 1. note 2.
• (week 12) heat kernel on Riemannian manifolds (continued), Faber-Krahn inequality and symmetrization argument. Reference: [3a] §3.2. [3c] ch. IV. note 2. note 3.
• (week 13) Cheeger's inequality, heat kernel and isoperimetric estimator. Reference: [3c] ch. IV and ch. V. note 3. note 4
• (week 14) heat kernel and embedding into l^2: construction, and small time limit. Reference: [3b] note 5.
• (week 15) heat kernel and embedding into l^2: choice of eigenbasis, isometric theorem. Weyl's law. Reference: [3b] note 5.

Homework

• Homework 1: due March 3.
• Homework 2: due March 10.
• Homework 3: due March 17.
• Homework 4: due March 24.
• Homework 5: due March 31.
• Homework 6: due April 7.
• Homework 7: due April 28.
• Homework 8: due May 5.