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1. Tuesday Lecture room 5507: (14:00-15:35);Thursday Lecture room 5507: (07:50-09:25).

Lecture 1 Introduction: Gauss 1827 and Riemann 1854; Smooth manifolds.

Lecture 2 Lengths of curves: Tangent spaces, Riemannian metric, tensors, distance function; Metric structure of Riemannian manifolds.

Lecture 3 Geodesics: looking for shortest curves: Examples: Spheres; geodesic equations and Christoffel symbols.

Lecture 4 Homogeneity of a geodesics; Exponential map and normal coordinates; Riemannian polar coordinates; Existence and uniqueness of local shortest curves.

Lecture 5 Totally normal neighborhood; Cut points and cut locus.

Lecture 6 Existence of shortest curves connecting any two points; Hopf-Rinow Theorem; Cut points revisited.

Lecture 7 Local isometry; Riemannian covering maps and completeness.

Lecture 8 Local isometries from complete Riemannian manifolds are Riemannian covering maps; Existence of shortest geodesics in a given homotopy class.

Lecture 9 Existence of closed geodesics in a given free homotopy class; Affine connections; Parallelism.

Lecture 10 Covariant derivatives of a general tensor field; Levi-Civita connection; The fundamental theorem in Riemannian geometry.

Lecture 11 Koszul formula; Geometric meaning of metric compatibility and torsion-freeness; First variation formula: Geodesic equation and Gauss lemma revisited.

Lecture 12 Second variation formula; Curvature tensor and Parallel transport; Riemannian manifolds with zero curvature tensor (I).

Lecture 13 Riemannian manifolds with zero curvature tensor (II); Covariant differentiation, Hessian, and Ricci identity.

Lecture 14 Ricci identity: local version; How Rieman discovered the curvature tensor; Bianchi identity; Riemannian curvature tensor and symmetry.

Lecture 15 Vanishing Riemannian curvature tensor appeared as an integrability condition; Sectional curvature; Second Bianchi identity and Schur's Theorem.

Lecture 16 Taking trace: divergence and Laplacian; Ricci curvature tensor and scalar curvature; Second Bianchi identity, Einstein tensor and Schur's Theorem; Ricci curvature and Bochner formula.

Lecture 17 Bakry-Emery Gamma Calculus; Second Variation Formula revisited; Bonnet-Myers theorem.

Lecture 18 Synge Theorem; Geodesics in space forms: Jacobi fields.

Homework 1.

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丁雁龙