I am a math enthusiast in electrical engineering and I am planning on learning Differential Geometry for applications in Control Theory. I want to teach myself this beautiful branch of mathematics in a rigorous way.
I am currently going through Chapman Pugh's Real Analysis, I am then planning on studying Munkres for Topology but I would have liked some advice to start out in DG. I was told Lee's Smooth Manifolds would be a nice, though tough, read. What do you think?
Best Answer
I taught myself Differential Geometry so I can tell you everything it's needed. First of all you will have to decide if go for classic differential geometry or calculus on manifold. I would suggest Calculus on Manifold since with a little bit of effort you will gain a lot. Having said that there's one secret to learn Differential Geometry, the secret that everybody knows and nobody does until they finally get illuminated: doing exercises. So the bad news are that studying the theory you will definetely have to work out a lot of exercises, the good news are that in general the exercises don't have to be very complicated to understand what's going on.
So I think your main book should be this one with exercises, answer and solutions that you need:
Selected Problems in Differential Geometry and Topology, by A.T. Fomenko, A.S. Mishchenko and Yu.P. Solovyev
Then there are a lot of good books which explain the theory, I would suggest a book that is easy to begin with as
Loring W. Tu, An Introduction to Manifolds (has also exercises with hints and solutions)
Then I think you can go for the classics Spivak, Do Carmo, Boothby and at that time you will be ready for Riemannian Geometry and you will be able to approach Nomizu or whatever book you like.