MAT5343

MAT 5343 Differential Geometry

Sample textbook: Victor V. Prasolov: Differential Geometry, Springer (2022)

Catalog entry:

Prerequisite: A course in Linear Algebra and Calculus sequence

Contents: This course offers an in-depth study of differential geometry, focusing on the geometric properties of differentiable manifolds. Topics include smooth manifolds, vector fields, differential forms, tensors, connections, Riemannian metrics, and curvature. Emphasis will be placed on understanding the interplay between geometry and topology, as well as applications in physics, particularly in general relativity.

Topics List

Week	Topic	Chapter from textbook	SLO
1	Curves in space and on the plane		Understand how to compute the curvature of a space/planar curve
2	Moving frames, Serret-Frenet formulas	2.1-2.5	Understand how apply the Serret-Frenet formulas, determine curve from given curvature and torsion
3	Surfaces in space, introduction to differential forms	3.1-3.4	Parameterize surfaces in space, compute and interpret first and second fundamental form
4	Gaussian curvature	3.5-3.7	Compute and interpret Gaussian curvature, understand Theorema Egregium.
5	Covariant differentiation on surfaces and parallel transport	3.8-3.11	Become fluent with tensor calculus and covariant differentiation
6	Gauss-Codazzi equations and Riemann curvature tensor	3.13-3.14	Be able to compute Riemann curvature tensor, become familiar with symmetry properties of the Riemann tensor
7	Riemannian manifolds I	5.1-5.5	Understand definition of Riemannian manifolds
8	Riemannian manifolds II	5.6-5.14	Understand properties of Riemannian manifolds
9	Introduction into Lorentzian manifolds and relativity	instructor provided material	Understand the difference between Riemannian and Lorentzian manifolds
10	Einstein manifolds	instructor provided material	Understand where Einstein equations come from
11	Global theorems in differential geometry	3.7, 5.4	Hopf-Rinow and Gauss-Bonnet theorem
12	Differential forms and integral theorems	instructor provided material	Be able to apply Stokes' theorem
13	Introduction to Lie groups	6.1-6.5	Be able to compute generators of a Lie group
14	Auxillary topics