Geometry and Physics

Question

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics.

The intended audience is Undergraduate Seniors who have experience with Algebra (Linear/Abstract), Point Set Topology and everything that is usually taken before these courses but no more topics like differential geometry.

The presentation can lasts 30 minutes.

I want to make it more accessible, so it has to be broad, and not specialized. e.g. history, fields, major contributions etc are more welcome than specific theorems or examples. Of course, I would love many examples but I will focus on them at introductory level.

I am looking for topics but more than that, resources.

Answer 1

Chapter 1 of John Lee's Introduction to Topological Manifolds talks about how manifolds are used in classical mechanics, general relativity and string theory to whet your appetite for more. It is available as a free sample.

Answer 2

If you have access to journals through your library, the "Journal of Geometry and Physics" may give a good insight and examples.

A couple of potentially useful book resources are "Mathematical Physics, Analysis and Geometry" (deMonvel and McKean) and "The Geometry of Physics: An Introduction" (Frankell, 2004).

I hope this helps

Answer 3

I'm not sure if this will help but I'll suggest some books that uses Differential Geometry in Physics. Of course, I'm not suggesting you to read them completely, it would take a lot of time, so the idea is that you just take a look on the books, see the main subjects being tackled and them see what they say about those specific subjects that you are interested in. From there you can have ideas on how to build your presentation.

The classic example of how Differential Geometry can be used in Physics is certainly General Relaitivity. So you talk a little bit about that, perhaps if you don't want to go into theoretic details, you can talk just a little bit about the history of how General Relativity developed needing Differential Geometry to succesfully model gravitation. A good place to look at this and see interesting results is the book "Semi-Riemannian Geometry" by Barret O'neill.

Also there's Michael Spivak "Physics for Mathematicians: Mechanics I". This book one part devoted to the Lagrangian Mechanics and another part devoted to Hamiltonian Mechanics. Since Spivak assumes the reader knows Differential Geometry he uses many constructions of manifolds to develop those ideas. You can take a look on the content there to see if you get good ideas to relate geometry to mechanics in your presentation.

Finally there's a book by the Brazilian Physicist Waldyr Alves Rodrigues Jr that talks about relationships between Maxwell, Einstein and Dirac equations. This book uses many differential geometry constructions to relate those equations. The book is called "The Many Faces of Maxwell, Dirac and Einstein equations."

These of course are places that have in depth treatment of the subjects you are interested to present. It would require you to search inside the books to see which subjects you want to present and how deep you want to go.

I hope this helps you somehow. Good luck with you presentation!