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1. Lecture room 5401: 2(15:55-18:20); 5(14:00-15:35).

Lecture 1 Brief review of Gauss 1827 and Riemann 1854; Smooth manifolds; Riemannian metric tensor.

Lecture 2 Existence of Riemannian metric; Pull-back metric; Product metric; Length of curves and distance function.

Lecture 3 Metric structure; Looking for shortest curves; Geodesic equations.

Lecture 4 Exponential map and normal coordinates; Riemannian polar coordinates; local existence and uniqueness of shortest curves.

Lecture 5 Totally normal neighborhood; Cut points and cut locus; Existence of the shortest curve between two points.

Lecture 6 Hopf-Rinow Thoerem; Existence of shortest curves in given homotopy classes.

Lecture 7 Riemannian covering map; Affine connections: an introduction.

Lecture 8 Affine connections: Locality and existence; Comparison with Lie derivative; Covariant derivative of smooth vector fields along a curve.

Lecture 9 Induced connections; Parallelism; Covariant derivatives of tensor fields; What is special for the connection determined by Christoffel symbols?

Lecture 10 Levi-Civita connections; First and second variation formulae of energy functional.

Lecture 11 Gauss lemma revisited; Curvature tensor and its geometric intuition; Riemann's original idea; Covariant differentiation.

Lecture 12 Ricci identity; Hessian, gradient, divergence and Laplacian.

Lecture 13 Divergence: Lie derivative of volume form; Riemannian manifolds with zero curvature tensor are locally isometric to Euclidean spaces; Bianchi identities.

Lecture 14 Sectional curvature; Schur Theorem.

Lecture 15 Ricci curvature tensor and Schur Theorem; Scalar curvature; Bochner identity. note

Lecture 16 Bakry-Emery calculus; Seconde variation formula revisited; Synge Theorem; Bonnet-Myers Theorem.

Lecture 17 Jacobi field (by Prof. Jianqing YU).

Lecture 18 Existence of simply connected space forms; Length of geodesic circles.

Lecture 19 Jacobi fields as critical points of the index form; Conjugate points and their characterization as critical points of exponential map; Piecewise smooth variations and index form; Algebraic properties of index form and occurancy of conjugate points (I).

    Midterm Exam (Lecture 1-16)

Lecture 20 Algebraic properties of index form and occurancy of conjugate points (II); Morse Index Theorem: Reduction to finite dimensional spaces.

Lecture 21 Morse Index Theorem: completion of the proof; Cartan-Hadamard Theorem.

Lecture 22 Uniqueness of simply-connected space forms; Convex functions.

Lecture 23 Convexity of distance functions on Cartan-Hadamard manifolds; Convex neighborhoods; Cut points revisited.

Lecture 24 Structure of complete Riemannian manifolds; Sturm's comparison theorem and its geometric translation due to Bonnet; Extensions to higher dimensional manifolds: Morse-Schoenberg comparison theorem.

Lecture 25 Rauch comparison theorem and its applications.

Lecture 26 Hessian and Laplacian comparison; Riemannian volume measure.

Lecture 27 Bishop-Gromov volume comparison theorem.

Lecture 28 Applications of relative volume comparison; Splitting theorem: Buseman function

Lecture 29 Splitting theorem: totally geodesic submanifolds and the proof. notes

Lecture 30 Topological sphere theorem: Klingenberg's injectivity radius estimates

Lecture 31 Topological sphere theorem: Berger and Tsukamoto's proof notes

Homework 1

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