MATH-GA 2360 Differential Geometry II

Thursdays 1:25-3:15pm, CIWW 201
Deane Yang
Recommended Books
Syllabus (Under construction)
Lecture Date Topics
1 1/25/2018 Hypersurface in Euclidean space:
Quick review of linear algebra
Implicit function theorem
Submanifold in Euclidean space
Tangent, cotangent bundles of submanifold
Induced metric, second fundamental form of hypersurface
Induced connection
Intrinsic curvature
Poincar\'e lemma
Frobenius theorem
Fundamental theorem of hypersurfaces in Euclidean space
2 2/1/2018 Intrinsic geometry of a manifold:
Intrinsic definition of manifold
Intrinsic definitions of tangent, cotangent, and tensor bundles
Intrinsic definition of metric, connection, and curvature
Examples: Sphere, hyperbolic surface (sine-Gordon)
Quick intro to tensors
Riemann and Ricci curvature tensors, scalar curvature
3 2/8/2018 Matrix groups:
Basic examples
Push-forward of vector field
Left invariant vector fields and Lie algebra
Lie bracket and Maurer-Cartan equations
Group of rigid motions as subgroup of GL(n+1)
Left invariant differential forms and Maurer-Cartan
Moving frames on hypersurface in Euclidean space
Frobenius theorem
Fundamental theorem of hypersurfaces
4 2/15/2018 Moving frames:
Sphere, projective spaces, Grassmannians
Moving frames on Riemannian manifold
Moving frame approach to stereographic projection
Principal bundle formulation
5 2/22/2018 Geodesics:
Energy functional
Variational formulas, Jacobi fields
Parallel transport and holonomy Geodesics and distance Jacobi equation, Sturm comparison theorem
Applications of comparison theorem
Closed geodesics and fundamental group
6 3/1/2018 Geometric consequences of Ricci curvature bounds
Laplacians
Bochner formulas
Volume comparison, proof of Bishop-Gromov
Isoperimetric inequalities
7 3/8/2018 PDEs on a Riemannian manifold
Sobolev norms and inequalities
Linear elliptic and parabolic PDEs on a Riemannian manifold
Hodge theorem
  3/15/2018 Spring Recess
8 3/22/2018 Nonlinear PDEs on a Riemannian manifold
9 3/29/2018 Ricci curvature equations
DeTurck trick
Prescribed Ricci equation
Einstein equation
Ricci flow
10 4/5/2018 TBA
11 4/12/2018 TBA
12 4/19/2018 TBA
13 4/26/2018 TBA
14 5/3/2018 TBA