Teaching-Liu


Home

Activities

Publications

Talks

Teaching

Research Seminars

Riemannian Geometry

(Spring 2022)


Notices

2.Final exam: Room 5503: 2020.06.14 14:30-16:30.

1. Lecture room 5403: 2(15:55-18:20); 5(14:00-15:35).


Lectures and excercises

Lecture 1 Introduction; Manifolds; Tangent spaces; Riemannian metric.

Lecture 2 Tensors; Existence of Riemannian metric; Distance function and metric structure. Explanation on paracompactness and the concept of tensors

Lecture 3 Looking for shortest curves in Eulidean and Spherical geometry; Energy functional; geodesic equation.

Lecture 4 Looking for charts with simple geodesic equations: exponential map. Why do we choose Christoffel symbols to be symmetric?

Lecture 5 Local existence and uniqueness of shortest curves; Totally normal neighborhood; Cut point.

Lecture 6 Existence of shortest curves between any two points: Completeness.

Lecture 7 Hopf-Rinow Theorem; Existence of closed geodesics.

Lecture 8 Existence of shortest curves in a given homotopy class; Riemannian covering maps.

Lecture 9 Affine connections; Vector fields along a curve; Parallelism.

Lecture 10 Covariant derivatives of tensor fields; Affine connections determined by Christoffel symbols.

Lecture 11 Levi-Civita connection; First and second variation formulae; Curvature tensor.

Lecture 12 Curvature tensor: Integrability condition; Covariant differentiation and Ricci identity.

Lecture 13 Local version of Ricci identity; Hessian, divergence and gradient.

blackboard note pdf, png

Lecture 14 Laplacian; Bianchi identities; Symmetries of Riemannian curvature tensor.

blackboard note pdf, png

Lecture 15 Sectoional, Ricci and scalar curvatures; Schur Theorem.

blackboard note pdf, png

Lecture 16 Bochner identity; Second Variation Formula revisited; Synge Theorem.

Lecture 17 Bonnet-Myers Theorem; Jacobi field; Length of geodesic circles.

Lecture 18 Jacobi field and conjugate points.

Lecture 19 Index forms.

Lecture 20 Index Lemma; Morse Index Theorem; Cartan-Hadamard Theorem

Lecture 21 Uniqueness of simply connected space form: non-positive curvature.

Lecture 22 Uniqueness of simply connected space form: positive curvature; Convexity; Cut points revisited.

Lecture 23 Continuity of cut locus; Sturm comparison theorem and its geometric translations by Bonnet.

Lecture 24 Transport a vector filed along a curve to another space; Morse-Schoenberg comparison theorem; Rauch comparison theorem; Sum of the angles of a geodesic triangle in nonpositively curved manifolds.

Lecture 25 Hessian comparison theorem.

Lecture 26 Laplacian comparison theorem; volume density and Jacobi fields.

Lecture 27 Volume and Riemannian measure; Relating volume to Laplacian.

Lecture 28 Bishop-Gromov volume comparison theorem; Cheng's Maximal diameter theorem; Complete Riemannian manifolds with nonnegative Ricci curvature: Yau's volume growth rate estimate; Busemann functions.

Lecture 29 Cheeger-Gromoll splitting Theorem (I): The Buseman function is harmonic with parallel gradient field.

Lecture 30 Cheeger-Gromoll splitting Theorem (II): Totally geodesic submanifolds and the isometric map.

Homework 1 Christoffel symbols and Geodesics.

Homework 2 Connections and Curvatures.

Homework 3 Jacobi fields.

Lecture notes by Jun Hao Tian What is a "nice" metric?.


Tutorials

助教

田珺昊 tian18@mail.ustc.edu.cn

黄文瑞 hwr24@mail.ustc.edu.cn


Geometric analysis on graphs 2021 Fall


Riemannian Geometry 2021 Spring


Differential Geometry 2020 Fall


Riemannian Geometry 2020 Spring


2017级华罗庚讨论班2019-2020


Differential Geometry 2019 Fall


Riemannian Geometry 2019 Spring


Differential Geometry 2018 Fall


纯粹数学前沿 2018 夏季学期


Riemannian Geometry 2018 Spring


Differential Geometry 2017 Fall


Riemannian Geometry 2017 Spring


Teaching in Durham