MATH5061 - Riemannian Geometry I - 2023/24

Announcement

• The final written examination will take place in-class on Apr 22 (Mon), 2024 from 2:30PM - 5:00PM.
• Problem Set 5 (the last one) is posted and due on Apr 15.
• Problem Set 4 is posted and due on Mar 25.
• Problem Set 3 is posted and due on Mar 11.
• Problem Set 2 is posted and due on Feb 26.
• The lecture on Jan 29 will take place at LSB 219.
• Problem Set 1 is posted and due on Feb 5.
• The first class will be on Jan 8, 2024. Note that the class on Jan 15 will be cancelled.

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General Information

Lecturer

• LI Man-chun Martin
  • Office: LSB 236
  • Tel: 3943-1851
  • Email:
  • Office Hours: By appointment

Time and Venue

• Lecture: Mon 2:30PM-5:15PM at LSB 222

Course Description

This course is intended to provide a solid background in Riemannian Geometry. Topics include: affine connection, tensor calculus, Riemannian metric, geodesics, curvature tensor, completeness and some global theory. Students taking this course are expected to have knowledge in differential geometry of curves and surfaces.

References

• "Lectures on Differential Geometry" by S.S. Chern, W.H. Chen & K.S. Lam
• "Riemannian Geometry" by M. do Carmo
• "Riemannian Geometry" by S. Gallot, D. Hulin & J. Lafontaine
• "Riemannian Geometry and Geometric Analysis" by J. Jost
• "Riemannian Manifolds: An Introduction to Curvature" by J. M. Lee

Pre-class Notes

• Course outline

Lecture Notes

• L1 - Abstract manifolds and submanifolds
• L2 - Tangent bundles and vector fields
• L3 - Tensors and Lie derivative
• L4 - Differential forms
• L5 - Riemannian metrics and connection
• L6 - Geodesics and exponential map
• L7 - Riemann Curvature Tensor
• L8 - Submanifold theory
• L9 - First and second variation formula for curves
• L10 - Jacobi fields
• L11 - Conjugate points and minimizing geodesics
• L12 - Completeness of Riemannian manifolds
• L13 - Bonnet Myers and Cartan Hadamard Theorem
• L14 - Space forms
• L15 - Comparison theorems in Riemannian geometry

Assignments

• Problem Set 1 (due on Feb 5)
• Problem Set 2 (due on Feb 26)
• Problem Set 3 (due on Mar 11)
• Problem Set 4 (due on Mar 25)
• Problem Set 5 (due on Apr 15)

Solutions

• Solutions to Assignment 1
• Solutions to Assignment 2
• Solutions to Assignment 3
• Solutions to Assignment 4
• Solutions to Assignment 5

Assessment Scheme

Homework Assignments		50%
Final Examination (in-class on Apr 22, 2024)		50%

Honesty in Academic Work

The Chinese University of Hong Kong places very high importance on honesty in academic work submitted by students, and adopts a policy of zero tolerance on cheating and plagiarism. Any related offence will lead to disciplinary action including termination of studies at the University. Although cases of cheating or plagiarism are rare at the University, everyone should make himself / herself familiar with the content of the following website:

http://www.cuhk.edu.hk/policy/academichonesty/

and thereby help avoid any practice that would not be acceptable.

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Assessment Policy

Last updated: April 08, 2024 23:14:04