Roadmap for learning symplectic geometry, starting from differential geometry

Question

I am a Physics Major. I have done an undergraduate level course of differential geometry. I want to get into symplectic and Riemannian geometry, but I would like to start over from differential one again. I found this book that seems to cover what I want, but I am not sure if it is a good book or not. Can someone recommend a book or other sources (or even roadmap of multiple books) that start from differential geometry and teaches till (or gives proper introduction to) Riemannian and symplectic geometry (physics perspective is preferred but not necessary).

Answer 1

If you have what it takes, Abraham-Marsden "Foundations of Mechanics" is a classic and one of the best books (if not the best) on symplectic geometry aimed to mechanics. The first two chapters cover the programme of a course in differential geometry. Everything is done formally but beware it require a certain degree of mathematical maturity.

Another book you may try is Frankel's "the Geometry of Physics". This one is much easier to read and an excellent book, the problem is that is less coincise.

Answer 2

I have self-studied a lot of symplectic geometry too - so here are the ones I found most helpful:

Lectures notes by da Silva, which also does a really good job at explaining other related geometries, such as Kähler.

Mechanics and Symmetry by Marsden, which also considers some infinite dimensional cases of symplectic manifolds.

Introduction to Symplectic Geometry by Rolf Berndt, which is a very detailed, but rather short textbook.

Introduction to Symplectic Topology, which is quite long and covers many topic, but many of them are of special interest. If you arent really interested in topological invariants, this book is probably not your choice.

Once you have got a decent grasp of either of these texts, just pick your problem and fill in the gaps that you have using the other ones. For Riemannian geometry, you can read through portions of

Connections, Curvature and Characteris Classes by Tu, as already recommended.

Josts textbook on Riemannian geometry and geometric analysis is also a good starting points, if you are into PDE stuff.

Alternatively, just learn Riemannian geometry whenever it shows up.