Beijing Institute of Mathematical Sciences and Applications Beijing Institute of Mathematical Sciences and Applications

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About
President
Governance
Partner Institutions
Visit
People
Management
Faculty
Postdocs
Visiting Scholars
Administration
Academic Support
Research
Research Groups
Courses
Seminars
Join Us
Faculty
Postdocs
Students
Events
Conferences
Workshops
Forum
Life @ BIMSA
Accommodation
Transportation
Facilities
Tour
News
News
Announcement
Downloads
Qiuzhen College, Tsinghua University
Yau Mathematical Sciences Center, Tsinghua University (YMSC)
Tsinghua Sanya International  Mathematics Forum (TSIMF)
Shanghai Institute for Mathematics and  Interdisciplinary Sciences (SIMIS)
BIMSA > Riemannian Geometry
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
Lynn Heller
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
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).
Beijing Institute of Mathematical Sciences and Applications
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Tel. 010-60661855 Tel. 010-60661855
Email. administration@bimsa.cn

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