Research Seminars

Riemannian Geometry

(Spring 2022)


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?.



田珺昊 tian18@mail.ustc.edu.cn

黄文瑞 hwr24@mail.ustc.edu.cn

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