MATH-GA 2360 Differential Geometry II

Wednesdays, 3:20pm - 5:10pm, starting February 7
Deane Yang
Recommended Books
Syllabus (always evolving)

The course will begin with a quick review of some topics already covered during the fall, including the proofs of some theorems presented then. After that, the course will focus on advanced topics, centered around the use of PDEs in Riemannian geometry. Possible topics include fundamental results on elliptic and parabolic PDEs on a Riemannian manifold, the Hodge theorem, the Ricci flow, isoperimetric and Sobolev inequalities.

Lecture Date Topics
1 2/7/2018
  • Useful technical theorems
    • Implicit function theorem
    • Constant rank theorem
    • Poincar\'e lemma
    • Frobenius theorem
  • Submanifolds of Euclidean space
    • Submanifold in Euclidean space
      • As embedding
      • As level set of submersion
    • Tangent, cotangent bundles of submanifold
    • Metric, connection, second fundamental form
    • Intrinsic curvature
    • Moving frames
2 2/14/2018
  • Quick review of Riemannian geometry
    • Covariant differentiation and curvature of a $1$-parameter family of curves
    • Length and energy of curves
    • Length space strucgture of a Riemannian manifold
    • Geodesics
    • Hopf-Rinow theorem
    • First and second variation of energy
    • Jacobi fields
    • Conjugate points
3 2/21/2018
  • Geodesics and Jacobi fields
    • Sturm comparison theorem
    • Global theorems on sectional curvature
4 2/28/2018 Guest lecture by Matt Grayson
5 3/7/2018
  3/14/2018 Spring Recess
6 3/21/2018
  • Geometric consequences of Ricci curvature bounds
    • Laplacian
    • Bochner formulas
    • Volume comparison, proof of Bishop-Gromov
  • Isoperimetric inequality
    • Eucldean space
    • Closed Riemannian manifold
7 3/28/2018
  • Elliptic and parabolic PDEs on a Riemannian manifold
    • Linear elliptic and parabolic PDEs
      • Hodge theorem
      • Heat equation
    • Nonlinear PDEs
      • Prescribed Ricci
      • Einstein
      • Ricci flow
8 4/4/2018
  • Ricci flow
9 4/11/2018 TBA
10 4/18/2018 TBA
11 4/25/2018 TBA
12 5/2/2018 TBA