2. Mid-term exam: 2021. 05.13 (Thursday): 16：00-18：00，Room: 5203

Lecture 1 Introduction; Smooth manifolds; Tangent spaces; Riemannian metric.

Lecture 2 Existence of Riemannnian metrics; Distance functions and metric structure; Energy functional and geodesic equations.

Lecture 3 Local existence and uniquess of geodesics; Homogenity; Introduction to existence and uniqueness of shortest curves: the cases of Euclidean plane and round sphere.

Lecture 4 Exponential Map; Normal neighborhood and Riemannian polar coordinates; Local existence and uniqueness of shortest curves; Totally normal neighborhood.

Lecture 5 Cut point; Existence of shortest curves between any two points; Hopf-Rinow Theorem (I).

Lecture 6 Hopf-Rinow Theorem (II); Cut locus and the structure of complete Riemannian manifolds.

Lecture 7 A brief introduction to conjugate points; Existence of closed geodesics.

Lecture 8 Existence of shortest curves in a given homotopy class; Riemannian covering maps; Affine connection: motivation, definition, locality and existence.

Lecture 9 Refined locality of affine connection; Covariant derivatives of vector fields along a curve; Parallelism.

Lecture 10 Covariant derivatives of tensor fields; Levi-Civita connection; First variation formula of energy functional.

Lecture 11 Second variation formula; Curvature tensor and its geometric intuition; Riemann's original idea.

Lecture 12 Riemann's original idea (II); Riemmannian manifolds with zero curvature tensor are locally isometric to Euclidean space; Covariant differentiation: Ricci identity, Hessian, gradient, divergence and Laplacian.

Lecture 13 First and second Bianchi identities; Symmetries of Riemannian curvature tensor; Sectional curvatures.

Lecture 14 Ricci curvature tensor and scalar curvature; Schur Theorem; Second Variation Formula Revisited; Synge Theorem (I).

Lecture 15 Synge Theorem (II): Orientability and Parallel transport; Bonnet-Myers Theorem.

Lecture 16 A brief review from Synge Theorem to results of Gromoll-Meyer, Cheeger-Gromoll and Perelman; Geodesics in Rⁿ, Sⁿ, Hⁿ; Jacobi fields.

Lecture 17 Jacobi fields: characterization as variational fields of geodesics; Tangential and normal Jacobi fields; Conjugate points and its characterization as critical points of exponential maps; Jacobi fields as critical points of the index form.

Lecture 18 Index form: definition, relation with Jacobi equation and second variation of energy functional; Algebraic properties of index form and occurency of conjugate points along a geodesic.

Lecture 19 The index lemma; Morse index theorem.

Lecture 20 Cartan-Hadamard theorem; Uniqueness of simply connected space forms: The case of nonpositive curvature.

Lecture 21 Uniqueness of simply connected space forms: the case of positive curvature; Convex functions and convex sets in Riemannian manifolds.

Lecture 22 Sturm comparison theorem and its geometric translations. Index comparison theorem.

Lecture 23 Morse and Schoenberg comparison theorem; Rauch comparison theorem and applications; Hessian comparison theorem (I).

Lecture 24 Hessian comparison theorem (II); Laplacian comparison theorem.

Lecture 25 Rigidity in Laplacian comparison theorem; Volume comparison theorem (I).

Lecture 26 Volume comparison theorem (II). Applications: Cheng maximal diameter theorem and volume growth of complete Riemannian manifolds with nonnegative Ricci curvture.

Lecture 27 (By Keshu Zhou 周可树) Soul conjecture.

Lecture 28 Cheeger-Gromoll Splitting Theorem (I): Busemann function

Lecture 29 Cheeger-Gromoll Splitting Theorem (II): Bochner formula

Lecture 30 Topological sphere theorem (I): Cut locus and injectivity radius.

Lecture 30 Topological sphere theorem (II): Klingenberg, Berger, Tsukamoto.

Homework 1: Riemannian metrics and Energy functional

Homework 2: Geodesics

Homework 3: Connection, parallelism and covariatn derivatives

Homework 4: Curvature

Homework 5: Jacobi fields

助教

吴文静 anprin@mail.ustc.edu.cn

鲁志豪 lzh139@mail.ustc.edu.cn

习题课

时间地点待定