Recommendations for differential geometry textbooks that develop geometric intuition.

Question

I'm currently self-studying complex analysis (CA), and reading "Visual Complex Analysis" by Tristan Needham. I'm absolutely fascinated by how much geometric intuition he provides for the key findings in CA. It has been a very enticing read so far.

I have a mechanical engineering background, I've previously self-studied general/algebraic topology, and I'm interested in self-studying differential geometry (DG) after finishing Needham's book. I know that Needham is in the process of releasing his next book, "Visual Differential Geometry". But the exact date of release is hard to find. Can anyone recommend a few good DG textbooks that (a) pay special attention to developing the geometric intuition of the reader (and perhaps less attention to rigorous mathematical proofs), and (b) would be appropriate for a reader with my aforementioned background?

Answer 1

Well, Spivak's (what was it $5$?) volume Comprehensive Introduction to Differential Geometry is in my opinion pretty good. And it seems to me he makes some effort to develop geometric intuition.

Sorry it's been so long that I can't remember specific examples ( I want to say he did $\Bbb RP^2$ pretty well).

I just remember calling Book Scientific, to order some books, and wouldn't you know it, Spivak answered the phone.

The covers of the volumes, incidentally, fit together into one big picture.

I dare say they're a must. And when I arrived at UCLA from Berkeley, what did I see on my advisor's shelf, but Spivak's epic.

Answer 2

I am looking forward to Visual Differential Geometry too. On Tristan Needham's website there is currently no exact release date, but there is a full title and publisher:

His new book, Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts, will be published in 2020 by Princeton University Press.

(Update: Visual Differential Geometry was published in July 2021.)

In the meantime, if you want some exposure to classical differential geometry at an introductory level, you might try Pressley's Elementary Differential Geometry. (See the MAA website for a review by Gouvêa.) Or you might try Tapp's Differential Geometry of Curves and Surfaces. (See the MAA website for a review by Hunacek.)

Neither book teaches differential forms, manifolds, connections or the like. This could be a pro or a con, but if your focus is developing some feeling for the subject in 2D or 3D then it might be an advantage. On the other hand, if you do want a gentle introduction to differential forms, Fortney's recent A Visual Introduction to Differential Forms and Calculus on Manifolds could be worth a look or Bachman's A Geometric Approach to Differential Forms.

You say you've studied general and algebraic topology, so you could also go straight to a book on manifolds such as Tu's An Introduction to Manifolds in preparation for more advanced books on differential geometry. That said, I note that in a comment you mention computational meshes. Edelbrunner's Geometry and Topology for Mesh Generation might be worth checking out. See also the resources on discrete differential geometry at http://ddg.cs.columbia.edu/.

Answer 3

Although the book is to be available 14 July in the states, if you buy it directly from Princeton University Press, they'll ship it to you now.

I'm only through 55 of its 464 pages of text, and it's fabulous. Tons of pictures. The diagrams to explain the changes in the metric from the sphere to the complex plane are gems.

You cannot find a page with only text and no equations or pictures.

Buy it