C3.11 Riemannian Geometry (2024-25)
Main content blocks
- Lecturer: Profile: Andrew Dancer
Course information
General prerequisites:
Differentiable Manifolds is required. An understanding of covering spaces will be strongly recommended.
Course term: Hilary
Course lecture information: 16 lectures
Course weight: 1
Course level: M
Course overview:
Riemannian Geometry is the study of curved spaces and provides an important tool with diverse applications from group theory to general relativity. The surprising power of Riemannian Geometry is that we can use local information to derive global results.
This course will study the key notions in Riemannian Geometry: geodesics and curvature. Building on the theory of surfaces in R\(^3\) in the Geometry of Surfaces course, we will describe the notion of Riemannian submanifolds, and study Jacobi fields, which exhibit the interaction between geodesics and curvature.
We will prove the Hopf--Rinow theorem, which shows that various notions of completeness are equivalent on Riemannian manifolds, and classify the spaces with constant curvature.
The highlight of the course will be to see how curvature influences topology. We will see this by proving the Cartan--Hadamard theorem, Bonnet--Myers theorem and Synge's theorem.
This course will study the key notions in Riemannian Geometry: geodesics and curvature. Building on the theory of surfaces in R\(^3\) in the Geometry of Surfaces course, we will describe the notion of Riemannian submanifolds, and study Jacobi fields, which exhibit the interaction between geodesics and curvature.
We will prove the Hopf--Rinow theorem, which shows that various notions of completeness are equivalent on Riemannian manifolds, and classify the spaces with constant curvature.
The highlight of the course will be to see how curvature influences topology. We will see this by proving the Cartan--Hadamard theorem, Bonnet--Myers theorem and Synge's theorem.
Learning outcomes:
The candidate will have great familiarity working with Riemannian metrics, the Levi-Civita connection, geodesics and curvature, both in a local coordinate description and using coordinate-free expressions. The candidate will gain understanding of Riemannian submanifolds, Jacobi fields, completeness, and be able to prove and apply fundamental results in the subject, including the theorems of Hopf--Rinow, Cartan--Hadamard, Bonnet--Myers and Synge.
Course synopsis:
Riemannian manifolds: basic examples of Riemannian metrics, Levi-Civita connection.
Geodesics: definition, first variation formula, exponential map, minimizing properties of geodesics.
Curvature: Riemann curvature tensor, sectional curvature, Ricci curvature, scalar curvature.
Riemannian submanifolds: examples, second fundamental form, Gauss--Codazzi equations.
Jacobi fields: Jacobi equation, conjugate points.
Completeness: Hopf--Rinow and Cartan--Hadamard theorems
Constant curvature: classification of complete manifolds with constant curvature.
Second variation and applications: second variation formula, Bonnet--Myers and Synge's theorems.
Geodesics: definition, first variation formula, exponential map, minimizing properties of geodesics.
Curvature: Riemann curvature tensor, sectional curvature, Ricci curvature, scalar curvature.
Riemannian submanifolds: examples, second fundamental form, Gauss--Codazzi equations.
Jacobi fields: Jacobi equation, conjugate points.
Completeness: Hopf--Rinow and Cartan--Hadamard theorems
Constant curvature: classification of complete manifolds with constant curvature.
Second variation and applications: second variation formula, Bonnet--Myers and Synge's theorems.
Section outline
-
-
Opened: Saturday, 25 January 2025, 10:48 PMThis problem sheet is based on the material in Sections 1 and 2 of the lecture notes.
Sections A and B consist of core material and Section C consists of a slightly more advanced question.
Students should only hand in solutions to questions in Section B. -
Opened: Monday, 27 January 2025, 4:12 PMThis problem sheet is based on the material in Sections 2.3 and 3 of the lecture notes.
There are three sections to the problem sheet. Section A consists of core material, Section B consists of core material and some more supplementary questions and Section C consists of a more advanced question based on core material. All questions are definitely worth attempting, with Sections A and B the most important.
Students should NOT hand in any solutions. -
Opened: Wednesday, 5 February 2025, 5:56 PMThis problem sheet is based on the material in Sections 4, 5 and 6 of the lecture notes.
There are three sections to the problem sheet. Section A consists of core material, Section B consists of core material and some more supplementary questions and Section C consists of a significantly more advanced question but which is very interesting and relevant. The questions from Sections A and B are important to do, and Section C's question is worth having a go at, especially as there is a big hint.
Students should only hand in solutions to questions in Section B. -
Opened: Monday, 17 February 2025, 3:28 PMThis problem sheet is based on the material in Sections 7, 8 and 9 of the lecture notes.
There are three sections to the problem sheet. Section A consists of core material and useful to do, Section B consists of core material at a slightly more advanced level and Section C consists of a slightly more advanced style of question which is definitely worth trying.
Students should NOT hand in solutions to questions. -
These lectures notes provide all of the essential material for the course, as well as some additional points for interest.
-
I give a list of what I consider to be essential formulae for the course. I also give a few useful but not essential formulae.
-
-
-
Registration start: Monday, 13 January 2025, 12:00 PMRegistration end: Friday, 14 February 2025, 12:00 PM
-
Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
-