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MA4C0 Differential Geometry

Lecturer: Shengwen Wang

Term(s): Term 1

Status for Mathematics students: List C

Commitment:

Assessment: 85% Examination, 15% Homework

Formal registration prerequisites: None

Assumed knowledge:

MA3H5 Manifolds: Key topics from MA3H5 will be covered rapidly in the first few lectures but you should do this module in Term 2 for a thorough discussion of them

MA251 Algebra I:

  • Bilinear forms
  • Eigenvalues and Eigenvectors

MA259 Multivariable Calculus:

  • Differentiation of functions of several variables, including the Chain Rule
  • Inverse and Implicit Function theorems

MA260 Norms, Metrics and Topologies OR MA222 Metric Spaces:

  • Basic point set topology

MA254 Theory of ODEs:

  • Existence and uniqueness of solutions to ODEs and their smooth dependence on parameters and initial conditions

Useful background:

Synergies:

Outline: The core of this course will be an introduction to Riemannian geometry - the study of Riemannian metrics on abstract manifolds. This is a classical subject, but is required knowledge for research in diverse areas of modern mathematics. We will try to present the material in order to prepare for the study of some of the other geometric structures one can put on manifolds.

Summary:

  • Review of basic notions on smooth manifolds; tensor fields
  • Riemannian metrics
  • Affine connections; Levi-Civita connection; parallel transport
  • Geodesics; exponential map; minimising properties of geodesics
  • The curvature tensor; sectional, Ricci and scalar curvatures
  • Training in making calculations: switching covariant derivatives; Bochner/Weitzenböck formula
  • Jacobi fields; geometric interpretation of curvature; second variation of length
  • Classical theorems in Riemannian Geometry: Bonnet-Myers, Hopf-Rinow and Cartan-Hadamard

Books:
Lee, J. M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, 1997
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer. 2nd edition, 1993
Jost, J.: Riemannian Geometry and Geometric Analysis 5th edition. Springer-Verlag, 2008
Petersen, P.: Riemannian Geometry Graduate Texts in Mathematics, 171. Springer-Verlag, 1998
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry
do Carmo, M: Riemannian Geometry. Birkhäuser, Boston, MA, 1992

Additional Resources