Intro to Differential Geometry 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?
 A: 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.
A: Arnold, do Carmo, and Spivak are very good books. Do stay away from Boothby. 
Guillemin and Pollack's very readable, very friendly introduction to topology is great, also Milnor's "Topology from the Differentiable Viewpoint". It will be useful to read them before or while you study the geometry part. 
I strongly recommend William Burke's Applied Differential Geometry. It's written in a conversational, intuitive style. Not every likes it. I think they are missing out on something.
Let me also mention Manifolds and Differential Geometry by Jeffrey M. Lee. It is quite complete, presenting manifolds, Lie groups, topology, forms, connections, and Riemannian geometry - probably has all one needs to know, and is much shorter that Spivak. It's an AMS book, thus good value for the money.
A: While the books in the comments and from the other answers are well and good, I would recommend that you also try out reading differential geometry text specifically written with engineering application in mind by an electrical engineer.
The book I am recommending is Shankar Sastry's Nonlinear System Analysis
The portion on differential geometry is written specifically for applications in control and written in a manner that most engineers can appreciate. 
I am not telling you to avoid the other texts, the ones by Loring Wu is particularly suitable for a first exposure. All I am trying to say is that over the decades engineers have advanced the field differential geometry to suit their own need, so if your end goal is control, why not start off reading what engineers had written for other engineers?
