Math 18.965: Geometry of Manifolds I

Tuesday, Thursday, 9:30-11am, 2-135

Course website: math.mit.edu/~tristanc/Math18.965.html

---

Instructor:

Tristan Collins

About the course

This course will cover basic notions of differential geometry, and geometric analysis.

Syllabus

The syllabus for the course is available here .

Course References

As much as possible, I will try to provide links to publicly available sources. Many textbooks are available on the web via the MIT Library.


• (doC) do Carmo: Riemannian Geometry
• (Pet) Petersen: Riemannian Geometry
• (ChEb) Cheeger and Ebin: Comparison Theorems in Riemannian Geometry
• (ScYa) Schoen and Yau: Lectures on Differential Geometry
• (Tau) Taubes: Differential Geometry
• (Hat) Hatcher: Algebraic Topology

---

Progress

This is approximate. I will attempt to keep track of all the references used in class.

Date	Material	Problems/Handouts
Sept. 5 --10	What is a manifold? References (doC, Chapter 0), (Tau, Chapter 1)	Homework 1
Sept. 12--17	Vector fields, the Tangent bundle, Lie derivative, metrics (doC. Chapter 0, 1)
Sept. 19- Sept. 26	Metrics, connections, geodesics: (doC, Chapter 1-3)	Homework 2
Sept. 26- Oct. 10	Tensors, Curvature, Jacobi Fields: (doC, Chapter 3-5 )	Homework 3
Oct. 15- 22	Covering spaces, theorems of Hopf-Rinow and Hadamard (doC, Chapter 7),(Hat, Chapter 1)	Homework 4
Oct. 24--Oct. 31	The cut locus, embedded submanifolds, mean curvature (doC, Chapter 6), (Pet, Sec. 5.7.3)
Nov. 5 -- Nov. 19	Space with constant curvature, comparison geometry (do C, Chapter 8), (Pet, Chapters 1, 3)	Homework 5
Nov. 21--Dec. 3	Bishop-Gromov, Cheng's sphere theorem, Cheeger-Gromoll splitting(Pet) (ScYa)	Homework 6
Dec. 3 - Dec. 9	The segment inequality, Cheng-Yau gradient estimate (Pet, Chapter 7) (ScYa)	Homework 7