1. First lecture: Feb. 25, 2020.

Lecture 1 Introduction; Riemannian metric: definition and existence. blackboard note

Lecture 2 Metric structure; Volume of a subset. blackboard note

Lecture 3 Riemannian measure; Geodesic equation. blackboard note

Lecture 4 Local existence and uniqueness of geodesics; Exponential map. blackboard note

Lecture 5 Geodesics and shortest curves. blackboard note, a correction

Lecture 6 Existence of shortest curves: from local to global blackboard note

Lecture 7 Hopf-Rinow Theorem; Injective radius; Local isometry.. blackboard note

Lecture 8 Existence of shortest geodesics in a given homotopy class. blackboard note

Lecture 9 Riemannian covering map; Laplace-Beltrami operator. blackboard note

Lecture 10 Sobolev space; Rellich compactness theorem; Poincare inequality. blackboard note

Lecture 11 Eigenvalues and eigenfunctions of Laplacian; Affine connection. blackboard note

Lecture 12 Existence and locality of affine connections; Induced connection. blackboard note

Lecture 13 Parallelism; Connections over tensor fields. blackboard note

Lecture 14 Levi-Civita connection. blackboard note

Lecture 15 First Variation; Covariant differentiation; Divergence. blackboard note

Lecture 16 Trace of Hessian is Laplacian; Ricci identity; Computation tricks. blackboard note

Lecture 17 Curvature tensor: its origin, geometric meaning and Bianchi identities. blackboard note

Lecture 18 Second Varation Formula; Riemannian curvature tensor. blackboard note

Lecture 19 Sectioinal, Ricci and Scalar curvature; Schur Theorem. blackboard note

Lecture 20 Ricci curvature and Bochner formula; Lichnerowicz estimate. blackboard note

Lecture 21 Second variation: revisited; Bonnet-Myers Theorem. blackboard note

Lecture 22 Synge Theorem; Geodesics in Rⁿ, Sⁿ, Hⁿ. blackboard note

Lecture 23 Jacobi filed: examples, characterizations, and decomposition. blackboard note

Lecture 24 Conjugate points; Index form as "Hessian" of Energy. blackboard note

Lecture 25 Deep relations between index forms and conjugate points. blackboard note

Lecture 26 Index of a geodesic is finite; Cartan-Hadamard Theorem. blackboard note

Lecture 27 Uniqueness of simply-connected space forms; Convex funtions. blackboard note

Lecture 28 Rauch comparison theorem; Cut point and cut locus. blackboard note, Examples and Illustrations

Lecture 29 Hessian and Laplacian comparison theorems. blackboard note

Lecture 30 Bishop-Gromov volume comparison theorem; Cheng's maximal diameter theorem. blackboard note

Homework 1: Differential Manifolds and Riemannian metrics Due: 2020/03/17

Homework 2: Geodesics Due: 2020/03/31

Lecture Notes: Function spaces and Laplace-Beltrami operator on Riemannian manifolds

Homework 3: Function spaces and Laplace-Beltrami operator Due: 2020/04/20

Homework 4: Connections, Parallelism, and Covariant derivatives Due: 2020/04/30

Homework 5: Curvature Due: 2020/05/21

Tutor

Wenbo Li (李文博) patlee@mail.ustc.edu.cn

Tutorials