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 1 5

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.

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