Riemannian Geometry
This is an 8-week introductory crash course on Riemannian Geometry. We begin with the definition of Riemannian manifolds and proceed to review several classical results, including the Bonnet–Myers theorem and various comparison theorems.
讲师
日期
2025年10月22日 至 12月12日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周三,周五 | 11:30 - 14:15 | A3-4-312 | ZOOM 05 | 293 812 9202 | BIMSA |
修课要求
Basic differential geometry (smooth manifolds, vector fields, differential forms), Familiarity with multivariable calculus and linear algebra
课程大纲
Week 1: Introduction to Riemannian Metrics
• Riemannian manifolds, examples, and basic constructions
• Tangent spaces and inner products
Week 2: Connections
• Levi-Civita connection
• Geodesics and parallel transport
Week 3: Curvature
• Riemann curvature tensor, sectional curvature, Ricci curvature
• Examples and computations
Week 4: Geodesic Completeness and the Exponential Map
• Hopf–Rinow theorem
• Distance function and completeness
Week 5: Comparison Theorems I
• Jacobi fields and conjugate points
• Second variantion of geodesics
Week 6: Comparison Theorems II
• Bonnet–Myers theorem
• Bishop–Gromov volume comparison
Week 7: Applications and Examples
• Spheres, hyperbolic spaces, and symmetric spaces
• Spaces of constant curvature
Week 8: Comparison theorems III
• Topological consequences of curvature bounds
• Riemannian manifolds, examples, and basic constructions
• Tangent spaces and inner products
Week 2: Connections
• Levi-Civita connection
• Geodesics and parallel transport
Week 3: Curvature
• Riemann curvature tensor, sectional curvature, Ricci curvature
• Examples and computations
Week 4: Geodesic Completeness and the Exponential Map
• Hopf–Rinow theorem
• Distance function and completeness
Week 5: Comparison Theorems I
• Jacobi fields and conjugate points
• Second variantion of geodesics
Week 6: Comparison Theorems II
• Bonnet–Myers theorem
• Bishop–Gromov volume comparison
Week 7: Applications and Examples
• Spheres, hyperbolic spaces, and symmetric spaces
• Spaces of constant curvature
Week 8: Comparison theorems III
• Topological consequences of curvature bounds
参考资料
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry
听众
Undergraduate
, Advanced Undergraduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Lynn Heller于2003-2008年在柏林工业大学学习经济学和在柏林工业大学学习数学，并于2012年在图宾根埃伯哈德卡尔斯大学获得博士学位。此后，她留在图宾根做博士后，直到2017年在汉诺威莱布尼茨大学获得初级教授职位。其在微分几何上有近 10 年的研究科研经历，特别是三维情况下的 constantmean-curvature (CMC)曲面和 constrained Willmore 曲面的微分几何问题，涵盖几何分析，可积系统，李代数，代数几何等多个领域。在国际重要期刊上发表论文 20 余篇，引用 100 余次，H 指数 6。在国际会议上受邀参与报告 20 余次。