The class meets twice a week, Tuesday/Friday, 5.30 - 7 pm. Room 105, Department of Mathematics.
Notes:
L1: Metric, Connection, geodesics, first variation
L2: Exponential maps, normal coordinates, Gauss lemma
L3: Hopf-Rinow, curvature tensor, Jacobi fields
L4: Second variation, Index form, Bonnet-Myers, Synge
L5: Submanifold geometry, Index lemmata (pathway to Heintze-Karcher)
L6: Rauch comparison, Cartan-Hadamard, metric and volume expansion in local coordinates
L7: Riccati, mean curvature, Laplace comparison, Bishop-Gromov
Upload 7': Hessian comparison, Convexity
L8, L9: Toponogov comparison, Sphere theorems: Toponogov-Cheng, Grove-Shiohama
L10, L11, extra upload: Gromov-Hausdorff, precompactness, applications, collapsing
L12, L13, L14, L15: Negatively curved spaces
L16, L17, L18: Heat kernel, stochastic completeness, Wiener measure and path integration
L17, L18, L19: Bochner identity, Lichnerowicz-Obata, Cheeger-Gromoll splitting, Cheeger constant, reverse Cheeger by Buser, Cheng-Yau gradient estimate