(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-sphereMR.
(5/08) the invariance of lambda, the criterion for homeomorphic spheres, the examples of Milnor. Reference: Milnor, On manifolds homeomorphic to the 7-sphereMR.
(5/13) calculation of lambda of the examples of Milnor. Reference: Milnor, On manifolds homeomorphic to the 7-sphereMR.
(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 manifoldMR. Rick Schoen and S.-T. Yau, Lectures on harmonic mapsMR.
(6/10) harmonic maps on surfaces, Hopf differential. Reference: Rick Schoen and S.-T. Yau, Lectures on harmonic mapsMR.