Difference between revisions of "MAT5343"
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| Line 16: | Line 16: | ||
| 1 | | 1 | ||
|| [[Curves in space and on the plane]] | || [[Curves in space and on the plane]] | ||
| − | || | + | || |
|| Understand how to compute the curvature of a space/planar curve | || Understand how to compute the curvature of a space/planar curve | ||
|- | |- | ||
| 2 | | 2 | ||
| − | || [[]] | + | || [[Moving frames, Serret-Frenet formulas]] |
| − | || 2 | + | || 2.1-2.5 |
| − | || | + | || Understand how apply the Serret-Frenet formulas, determine curve from given curvature and torsion |
|- | |- | ||
| 3 | | 3 | ||
| − | || [[ ]] | + | || [[Surfaces in space, introduction to differential forms]] |
| − | || 3 | + | || 3.1-3.4 |
| − | || | + | || Parameterize surfaces in space, compute and interpret first and second fundamental form |
|- | |- | ||
| 4 | | 4 | ||
| − | || [[ ]] | + | || [[Gaussian curvature]] |
| − | || | + | || 3.5-3.7 |
| − | || | + | || Compute and interpret Gaussian curvature |
|- | |- | ||
| 5 | | 5 | ||
| − | || [[ ]] | + | || [[Covariant differentiation on surfaces and parallel transport]] |
| − | || | + | || 3.8-3.11 |
|| | || | ||
|- | |- | ||
| 6 | | 6 | ||
| − | || [[ ]] | + | || [[Gauss-Codazzi equations and Riemann curvature tensor]] |
| − | || | + | || 3.13-3.14 |
|| | || | ||
|- | |- | ||
| 7 | | 7 | ||
| − | || [[ ]] | + | || [[Riemannian manifolds I]] |
| − | || | + | || 5.1-5.5 |
|| | || | ||
|- | |- | ||
| 8 | | 8 | ||
| − | || [[ ]] | + | || [[Riemannian manifolds II]] |
| − | || 6 | + | || 5.6-5.14 |
|| | || | ||
|- | |- | ||
| 9 | | 9 | ||
| − | || [[ ]] | + | || [[Introduction into Lorentzian manifolds and relativity]] |
| − | || | + | || instructor provided material |
|| | || | ||
|- | |- | ||
| 10 | | 10 | ||
| − | || [[ ]] | + | || [[Einstein manifolds]] |
| − | || | + | || instructor provided material |
|| | || | ||
|- | |- | ||
| 11 | | 11 | ||
| − | || [[ ]] | + | || [[Global theorems in differential geometry]] |
| − | || | + | || 3.7, 5.4 |
| − | || | + | || Hopf-Rinow and Gauss-Bonnet theorem |
|- | |- | ||
| 12 | | 12 | ||
| − | || [[ ]] | + | || [[Differential forms and integral theorems]] |
| − | || | + | || instructor provided material |
|| | || | ||
|- | |- | ||
| 13 | | 13 | ||
| − | || [[ ]] | + | || [[Introduction to Lie groups]] |
| − | || 6 | + | || 6.1-6.5 |
|| | || | ||
|- | |- | ||
| 14 | | 14 | ||
| − | || [[ ]] | + | || [[Auxillary topics]] |
|| | || | ||
|| | || | ||
|} | |} | ||
Revision as of 07:30, 24 January 2025
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 |
| 5 | Covariant differentiation on surfaces and parallel transport | 3.8-3.11 | |
| 6 | Gauss-Codazzi equations and Riemann curvature tensor | 3.13-3.14 | |
| 7 | Riemannian manifolds I | 5.1-5.5 | |
| 8 | Riemannian manifolds II | 5.6-5.14 | |
| 9 | Introduction into Lorentzian manifolds and relativity | instructor provided material | |
| 10 | Einstein manifolds | instructor provided material | |
| 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 | |
| 13 | Introduction to Lie groups | 6.1-6.5 | |
| 14 | Auxillary topics |
