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MA6C0 Differential Geometry

Lecturers: Shengwen Wang

Term(s): Term 1

Commitment: 30 lectures

Assessment: Oral Exam

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:

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

Archived Pages: 2012 2015 2017