PHYS 581: Differential Geometry for Physicists

last updated: 7.50 AM, Dec 13, 2021

what From the Aharonov-Bohm effect in quantum mechanics to the curved spacetime of General Relativity, differential geometry is everywhere in physics. In this course we will explore the basics of differential geometry from a hands-on physicist's perspective and develop the necessary tools to understand the main ideas of spacetime and General Relativity.

Keywords: Vectors, matrices, tensors, and coordinates; why you need a covariant derivative; what is a metric; geometry of 2D surfaces; differentiable manifolds; what is curvature; why are differential forms so useful; selected applications (participants can decide what will be covered): spacetime as a differentiable manifold, Maxwell's equations as differential forms, curvature in General Relativity, gauge theories

when lecture: Friday, 11am, Small 122
office hours: Monday, 10-11am, Small 235
contact Jens Boos (

Take-home final

Due: Wednesday, Dec 15, 12pm. Please either submit via email or leave it in the mailbox of the instructor.



  1. 2021-09-03: introduction; vectors, matrices, and tensors [notes-0.pdf]
  2. 2021-09-10: vectors, covectors, matrices, and tensors [notes-1.pdf, assignment-1-v1.pdf, assignment-1-v1.tex]
  3. 2021-09-17: frames and coordinates [notes-2.pdf, assignment-2-v2.pdf, assignment-2-v2.tex]
  4. 2021-09-24: coordinates and coordinate transformations [notes-3.pdf, assignment-3-v1.pdf, assignment-3-v1.tex]
  5. 2021-10-01: the metric tensor [notes-4.pdf, assignment-4-v1.pdf, assignment-4-v1.tex]
  6. 2021-10-08: the covariant derivative [notes-5.pdf, assignment-5-v2.pdf, assignment-5-v2.tex]
  7. 2021-10-15: the covariant derivative [notes-6.pdf, no assignment this week, enjoy your fall break!]
  8. 2021-10-22: the Levi-Civita connection [notes-7.pdf, assignment-6-v1.pdf, assignment-6-v1.tex]
  9. 2021-10-29: Riemannian curvature [notes-8.pdf, assignment-7-v1.pdf, assignment-7-v1.tex]
  10. 2021-11-05: differential forms [notes-9.pdf, assignment-8-v2.pdf, assignment-8-v2.tex]
  11. 2021-11-12: Hodge duality [notes-10.pdf, no assignment this week]
  12. 2021-11-19: the codifferential [notes-11.pdf, assignment-9-v2.pdf, assignment-9-v2.tex]
  13. 2021-11-26: no class (thanksgiving)
  14. 2021-12-03: geometry of 2D surfaces [notes-12.pdf, assignment-10-v3.pdf, assignment-10-v3.tex]
  15. 2021-12-10: Topological phases, Aharonov–Bohm effect, and curvature as differential forms [notes-13.pdf]

Additional reading

  1. TBA