# 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?

• V. Arnol'd's "Ordinary differential equations" (specifically, look up: "manifolds"), and then his "Mathematical methods of classical mechanics".
– avs
Nov 28, 2016 at 4:41
• The only book that gets you learn this hard subject is Spivak's 3 volumes DG. It is written at a high level. Nov 28, 2016 at 4:41
• @DeepSea Are you talking about Spivak's A comprehensive introduction to Differential Geometry? It is my understanding that they are $5$ volumes... (?) By the way I couldn't agree more with you in that case. =)
– user378947
Nov 28, 2016 at 5:02
• Its true. That's the only one, but the cost is high. You learn it with a professor and not by yourself. Its hard. Nov 28, 2016 at 5:06
• There's a Dover book by Kreyszig on differential geometry, and that's the same Kreyszig I believe as the one who writes the maths for engineers books. Nov 28, 2016 at 6:39

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.

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.

• I understand why you're saying to stay away from Boothby, but the part on Lie Groups and connection forms is really well done for an introductory book. He would need to go to Nomizu or Helgason to find something similar on those topics and definitely wouldn't suggest those book to a beginner.
– Dac0
Nov 28, 2016 at 18:47
• Can you elaborate why stay away with Boothby? Nov 28, 2016 at 20:33
• It's not a bad book. However it is rather formal and dry - not formal in a mathematical way, but kind of pointing the reader to the formulas, the mechanics of the calculations. May serve as a references later on. It is a bad way to start, doesn't build intuition, vision, or ideas on how the formalism can be used. Many smart control engineers have the book on their shelves and gave up rather quickly.
– Pait
Nov 29, 2016 at 13:04
• I added the mention of Lee's book. The coverage is great, and a smoother read than Nomizu and other classics.
– Pait
Nov 29, 2016 at 13:16

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?

• Why? Because, honestly, other books are better - both at building the intuition and at presenting the mathematical material. Loring Tu is good but doesn't do the Riemannian geometry part - which may be where the material becomes useful for engineers. So perhaps the shorter Guillemin & Pollack is an easier way to read about manifolds and forms.
– Pait
Nov 29, 2016 at 13:07