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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.

Homework 1.

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丁雁龙