D I F F E R E N T I A L G E O M E T R Y I I , S P R I N G 2 0 2 3

- 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%

- Homework 30%
- 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.

- (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 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.

Last modified: June 2, 2023.

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