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Teaching 2025

Riemannian Geometry

After a couple of weeks dedicated to discussing manifolds, bundles, flows, and such, I will first aim for the timeless hits of Riemannian geometry: Myers, Synge, 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 on 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. I 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 manifolds, 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 differential. Inverse mapping theorem and implicit 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-Lindelöf and the flow associated with 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).
  • Week 7: Gauss lemma, Hopf-Rinow, curvature tensor and its symmetries, sectional curvature, Ricci curvature, scalar curvature, Jacobi equation and fields as tangent vector fields to geodesic variations.
  • Week 8: Jacobi fields and differential of the exponential map, Hadamard-Cartan, calculation of the curvature of H^n, R^n and S^n, Jacobi fields in constant curvature, first variation formula (again) and second variation formula, Bonnet-Myers.
  • Week 9: Existence of closed minimizing geodesics in homotopy classes, Synge's theorem, Killing fields, Noether's theorem, Claireaux's theorem, unimodular Lie groups, existence of bi-invariant metrics on compact Lie groups, calculation of Levi-Civita connection, curvature tensor and sectional curvature of bi-invariant metrics on Lie groups, proof that semisimple Lie groups with bi-invariant metrics are compact and have surjective exponential map, discussion of Killing form with proof that if G is semisimple and compact then the negative of the Killing form is an Einstein metric. Discussion (off-topic) of corollaries of Cartan's closed group theorem.
  • Week 10: Index Lemma, bounds for distance to conjugate points, Rauch's theorem, volume growth of simply connected manifolds of negative curvature, quasi-isometries and Milnor-Svarc lemma, exponential growth for of pi_1 for closed manifolds of negative curvature.
  • Week 11: Bishop-Gromov, manifolds of sectional curvature less or equal to K are CAT(K) spaces. Convexity of distance functions in CAT(0) spaces, finite/compact groups have fixed points, geodesics at bounded distance are parallel, maximal abelian subgroups of pi1 of negatively curved manifolds are cyclic, flat strip theorem, convex projection.
  • Week 12: Boundary of a CAT(-1) space, Morse lemma, quasi-isometries induce boundary map, delta-hyperbolicity.
  • Week 13: Boundary map is quasi-conformal, proof of Mostow rigidity (for hyperbolic manifolds) à la Tukia.
  • Topics I intended to do but didn't: Ambrose-Hicks theorem, classification of constant curvature spaces (it would have needed 5 minutes but I didn't find them), holonomy, de Rahm's theorem,...