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.
Lecturer
Date
22nd October ~ 12th December, 2025
Location
Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|
Wednesday,Friday | 11:30 - 14:15 | A3-4-312 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Basic differential geometry (smooth manifolds, vector fields, differential forms), Familiarity with multivariable calculus and linear algebra
Syllabus
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
Reference
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry
Audience
Undergraduate
, Advanced Undergraduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Lynn Heller studied economics at the FU Berlin and Mathematics at TU Berlin from 2003-2007 and obtained her PhD in mathematics from Eberhard Karls University Tübingen in 2012. Before joining BIMSA she was juniorprofessor at Leibniz University in Hannover.
For the period 2025-2028 Lynn Heller is serving as a member of the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU).
For the period 2025-2028 Lynn Heller is serving as a member of the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU).