Riemannian Geometry IV

Epiphany 2014

Time and place: Tuesday and Friday, 12:00 CM107
Instructor: Anna Felikson
e-mail: anna dot felikson at durham dot ac dot uk
Office: CM 124; Phone: 334-4158
Office hours: Friday 13:00-14:00

Recommended books:

S.Gudmundsson, An Introduction to Riemannian Geometry, Lecture Notes, Lund University (2012).
J.Lee, Riemannian Manifolds, An Introduction to curvature, Springer (1997)
M.P. Do Carmo, Riemannian Geometry, Birkhäuser (1992)

Further reading:

F.Morgan, Riemannian Geometry, Jones and Bartlett Publishers, (1998)
T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs 149, AMS (1996)
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer (2004)
J. Gallier, Notes On Group Actions, Manifolds, Lie Groups and Lie Algebra (2005)
J. Gallier, Notes on Differential Geometry and Lie Groups (2014)

Preliminary course content (subject to change): introduction to Lie groups, different notions of curvature (Riemannian curvature tensor, sectional curvature, Ricci curvature, scalar curvature), the second variation formula of the length and Bonnet-Myers Theorem, Jacobi fields, Theorem of Cartan-Hadamard.

Schedule: --------- red: most important notions, blue: most important statements

Week 11: Short overview of the first term. Lie groups : left-invariant vector fields and Lie algebras. Lie group exponential map.

Week 12: Adjoint representation. Riemannian metrics on Lie groups. Invariant metric on homogenious spaces.

Week 13: Curvature: Riemannian curvature tensor, sectional curvature.

Week 14: Problems class (on Riemannian tensor and sectional curvature). Ricci curvature, scalar curvature.

Week 15: Bonnet-Myers Theorem. Second variation formula of the length.

Week 16: Jacobi fields: Jacobi fields and geodesic variations, Jacobi fields and conjugate points.

Week 17: Normal Jacobi fields. Geodesics are not minimal past conjugate points.

Week 18: Problems class (Jacobi fields and conjugate points). Theorem of Cartan-Hadamard.

Week 19: Curvature and geometry: cut locus, injectivity radius, Sphere theorem, spaces of constant curvature, comparison triangles, Alexandrov-Toponogov Theorem.

If you have any questions you are very welcome to ask (during the lectures, after a lecture, during office hours, in any other convinient time or via e-mail)!!!

Homeworks:

There will be weekly sets of exercises; stared questions to hand in on Fridays, weeks 13,15,17,19. -- (+/- notation used for marking)

Weeks 11-12: ---- Exercises ---- Hints ----
Weeks 13-14: ---- Exercises ---- Hints ----
Weeks 15-16: ---- Exercises ---- Hints ----
Weeks 17-18: ---- Exercises ---- Hints ----

Handouts:

Birds-eye view of the first term
Term 1: outline
Term 2: outline

Standard problems: ---- see here

Fun: ---- Hairy Ball Theorem in 1-munute video

Who is who: ---- Hausdorff, ---- Riemann, ---- Lie, ---- Levi-Civita, ---- Christoffel, ---- Jacobi, ---- Gauss, ---- Hopf, ---- Rinow, ---- Ricci, ---- Bonnet, ---- Myers, ---- Cartan, ---- Hadamard.