Riemannian geometry
After a couple of weeks dedicated to discuss manifolds, bundles, flows and such, I will first aim to the timeless hits of Riemannian geometry: Myers, Singe, Hadamard-Cartan and Bishop-Gromov. I will then focus on manifolds of non-positive curvature: boundary at infinity, fundamental group, and such. The end goal is to discuss some rigidity result. Depending how things go I will either aim for BCG or for the rank-rigidity theorem. In any case, symmetric spaces will appear all the time. In encourage everybody to follow Guy Casale's class on Lie groups.When it comes to stuff about smooth manifolds, I recommend Lee's book Introduction to Smooth manifolds... Or mine. Afterwards, for a long time, do Carmo's Riemannian geometry is a beautiful book. I will add further references down the line.
Here is what I have covered so far:
- Week 1 : Topological manifolds, smooth manifold, maximal atlas, dimension is well-defined, smooth maps between smooth manifolds, composition of smooth maps is smooth, submanifolds, submanifolds are smooth manifolds in their own right, implicit function theorem and examples of submanifolds (S^n, SL_nR, SO_n), Lie groups, getting manifolds gluing charts (RP^n, CP^n), manifolds as quotients as free proper actions (RP^n, T^n, SL^nZ\SL^nR), existence of partitions of unity.
- Week 2: Tangent space and differerential. Inverse mapping theorem and implicite function theorem (preimage of a regular value is a submanifold). Every manifold can be embedded into some R^n. Fiber bundles (Hopf fibration is not trivial), vector bundles, principal bundles (Hopf fibration and frame bundle), vector bundle construction lemma. Construction of the tangent bundle, Whitney sum, dual bundle (more about this next week).
- Week 3: Sections of bundles, Whitney approximation, sections defined on closed sets can be extended to global sections, existence of Riemannian metrics, differential forms and exterior derivative.
- Week 4: Lie bracket, Picard-Lindelof and the flow associated to a vector field, Lie derivative of vector fields, differential forms and tensor fields, Lie derivative of vector field is given by Lie bracket, flows commute if and only if Lie bracket of corresponding vector fields vanishes, proof of Frobenius theorem, and geometric argument to prove that the distribution of complex planes in S^3 is not integrable.
- Week 5: Connections, every vector bundle admits a connection, connection induced on dual bundle, Levi-Civita connection, parallel transport, example in S^2, geodesics via ode, local existence, geodesically complete, first variation formula (as Euler-Lagrange equation with a potential)
- Week 6: Exponential map, differential of the exponential map at 0, Lie bracket, existence of regular neighborhoods, quotient manifold theorem (for free actions of compact groups), geometry of SL_nR/SO_n via Killing form, symmetric spaces (homogeneous, geodesically complete, G/K).