Differential Geometry II, Spring 2015

Course Information

• Syllabus
• Information
• Ceiba
• Lectures: Wednesday, 16:30 ~ 17:20 and Friday 10:20 ~ 12:10 at Astro-Math 102
• Office hours: Tuesday 16:00 ~ 17:00 and Thursday 10:45 ~ 11:45, or by appointment, at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have two jokers: the lowest two grades will be discarded.
  • Midterm 30%
    The midterm will be a written exam. It will be held during the class of April 24 or May 1.
  • Final 35%
    The final will be an oral presentation on some topic.
  • Course participation 5%
• Course Assistant: Yu-Ching Sun, whose office is Astro-Math 445.
  • CA office hour: Thursday, 16:00 ~ 17:00.
• References:
  • [CE] Jeff Cheeger and David Ebin, Comparison theorems in Riemannian Geometry. MR
  • [W] Frank Warner, Foundations of differentiable manifolds and Lie groups. MR
  • [T] Clifford Taubes, Differential geometry. Bundles, connections, metrics and curvature. MR
  • [dC] Manfredo do Carmo, Riemannian Geometry. MR

Lecture summaries and references

• (2/25) Levi-Civita connection. Reference: [CE, §1.0] and [dC, §2.3]
• (3/4) the method of moving frame, Riemann curvature tensor. Reference: [CE, §1.4] and [dC, §4.2], see also note0304
• (3/6) basic properties of Riemann curvature tensor, first variation formula, Jacobi field. Reference: [CE, §1.1 and 1.4] and [dC, §4.2 and 5.1]
• (3/11) the coefficients of the Gaussian coordinate. Reference: [CE, §1.4] and [dC, §4.2]
• (3/13) notions of curvatures (sectional, Ricci, scalar), second variational formula, index lemma. Reference: [CE, §1.6 and 1.8] and [dC, §9.2 and 10.2]
• (3/18) index lemma and its application. Reference: [CE, §1.8] and [dC, §10.2]
• (3/20) Bonnet--Myers theorem, Rauch comparison theorem, crash course on covering spaces and fundamental groups. Reference: [CE, §1.9 and 1.10] and [dC, §9.3 and 10.2],
  see Bredon: Topology and Geometry, ch.III. MR and Vick: Homology theory, ch.4. MR for the covering spaces and fundamental groups
• (3/25) covering spaces and fundamental groups (continued), application of Bonnet--Myers theorem. Reference: [CE, §1.9], see note0325 for a brief introduction to tensor calculus
• (3/27) Cartan--Hadamard theorem, Cartan--Ambrose--Hicks theorem. Reference: [CE, §1.11 and 1.12] and [dC, §7.3 and 8.2]
• (4/01) and (4/03) Spring Break.
• (4/08) Hodge star and Laplace--Beltrami operator. Reference: [W, §4.1]
• (4/10) Hodge theorem (assuming regularity and compactness). Reference: [W, §4.2]
• (4/15) Sobolev space and some basic properties, Sobolev embedding. Reference: [W, §4.3]
• (4/17) properties of Sobolev space, Rellich lemma, elliptic operator and elliptic estimate (on the torus). Reference: [W, §4.3 and 4.4]
• (4/22) difference quotient, elliptic regularity (on the torus). Reference: [W, §4.3 and 4.4]
• (4/24) regularity and compactness of Laplace (on any Riemannian manifold). Reference: [W, §4.5]
• (4/29) introduction to the local index theorem and the heat kernel approach.
• (5/01) Midterm.
• (5/06) intersection pairing and signature. Reference: Milnor, On manifolds homeomorphic to the 7-sphere MR.
• (5/08) the invariance of lambda, the criterion for homeomorphic spheres, the examples of Milnor. Reference: Milnor, On manifolds homeomorphic to the 7-sphere MR.
• (5/13) calculation of lambda of the examples of Milnor. Reference: Milnor, On manifolds homeomorphic to the 7-sphere MR.
• (5/15) TIMS 2015 Mini-Course on Topology and Geometry of Ricci Solitons: link.
• (5/20) Bochner Laplacian. Reference: Peter Li, Geometric analysis. MR.
• (5/22) Bochner formula and its applitcations. Reference: Peter Li, Geometric analysis. MR.
• (5/27) Killing vector field on negatively curved manifold. Reference: Peter Li, Geometric analysis. MR.
• (5/29) Eigenvalue estimate, Laplacian and second fundamental form, eigenvalue and minimal embedding. Reference: Peter Li, Geometric analysis. MR.
• (6/03) Reilly's theorem. Reference: Robert Reilly, Applications of the Hessian operator in a Riemannian manifold. MR.
• (6/05) Reilly's proof of Aleksandrov theorem, introduction to harmonic maps. Reference: Robert Reilly, Applications of the Hessian operator in a Riemannian manifold MR. Rick Schoen and S.-T. Yau, Lectures on harmonic maps MR.
• (6/10) harmonic maps on surfaces, Hopf differential. Reference: Rick Schoen and S.-T. Yau, Lectures on harmonic maps MR.

Homework

• Homework 01: due March 11.
• Homework 02: due March 18.
• Homework 03: due March 25.
• Homework 04: due April 8.
• Homework 05: due April 15.
• Homework 06: due April 22.
• Homework 07: due May 6.
• Homework 08: due May 13.
• Homework 09: due May 20.