I am planning to study geometry (I know this is very vague, but I can't exactly explain what I mean, because I don't fully understand the distinction between various terms, which is the reason for this question).
I'm confused by titles of books. As I understand, there are books which have "Smooth Manifolds" in their title, such as Lee's book Introduction to Smooth Manifolds. On the other hand, there are books which contain "Differential Geometry" in their title, such as Spivak's book A Comprehensive Introduction to Differential Geometry.
Looking at the table of contents, I don't really understand the difference between books on "smooth manifolds" and books on "differential geometry". Could someone explain what the distinction is, and perhaps suggest what is the usual order of studying them? In other words, would you first study something which has "differential geometry" in the title, or "smooth manifolds"?
Similar question about difference between Riemannian geometry and differential geometry
Best Answer
Smooth manifolds are like the background things you work with: the base layer/ingredients of a cake (eggs, flour etc). Geometry is the study of things like lengths, angles etc on smooth manifolds (like extra ingredients on a cake: chocolate, vanilla etc, each type of geometry having a completely different flavor, and techniques).
The stuff in Lee's Introduction to Smooth Manifolds book is indeed very similar in topics to say Spivak's volume 1. Now, regarding the name, here's a part quoted directly from Lee's preface to Introduction to Smooth Manifolds:
If you just want to get a feeling for what the subjects are about, I think reading the prefaces of these various books (Lee has 3 books, Spivak has 5 volumes (the first two being most introductory/"important")) is a good start.
As an example other than Lee and Spivak, here is the preface (in full, because this text is harder to find) to Dieudonne's Treatise on Analysis, Vol IV, Chapter XX (20), titled "Principal Connections and Riemannian Geometry", describing roughly what Riemannian geometry is about, and some of its history:
Obviously you don't have to udnerstand everything here; just gain whatever nuggets of inspiration/information you can and move on (for now:)