Differential Geometry 2

Differential Geometry 2- Definitions Fest

So many definitions…

The book I am studying out of is John Lee’s Introduction to Smooth Manifolds (GTM for the math people) and it almost requires a companion text, John Lee’s Introduction to Topological Manifolds, or a mastery of a fair amount of Point-Set and Algebraic Topology as a reference. I am just finishing with the second chapter

In the preface, Lee mentions that “It is the bane of this subject that there are so many definitions that must be piled on top of one another before anything interesting can be said, much less proved.” He is not lying. There are a LOT of definitions and lemmas. They kind of pop in and out without, for now, much context. This is hard for me. I like to look at math as a story and it is easier for me to grok and play with theorems, proofs, and abstract constructions when I have an idea of what questions these things answer and what problems they solve. I’m being patient though. I’m treating this like it’s a Dostoyevsky novel where the first couple hundred pages are characterization and world building.

Still this is tough. There was one notion, that of covering spaces, which gets a few pages of write up in “Smooth” yet comprises the whole second half of “topological.” So it will take me some time to ground myself in the first book in order to prepare myself for the second. Math is not easy y’all.

But it is fun. Despite the rigor and the many times where my brain gives out and begs for netflix, this is quite an adventure. Let me tell you something about what I’m learning.

Chapter one is about defining what a Smooth Manifold is and simply put, it is a nice (Second countable basis (why?) and Hausdorff (cool)) topological space that looks locally Euclidean (3 Dimensional Euclidean Space (aka R^3) is like the regular space we are used to, but in math things can have any number of dimensions) with a smooth structure. We start with Topological Manifolds. One of the aims of this book is to define everything as intrinsic to the manifold without sneaking in properties of R^n so Lee uses a lot of maps between our abstract manifold and R^n. These maps are homeomorphisms and the are called coordinate charts. He then gives some examples of some cool topological manifolds like Spheres in integer valued (n) dimensions and Projective Space (it will hurt your head but its cool, take my word for it).

Then we define smooth structures. The notion of smooth comes from calculus and is means “infinitely differentiable” which in turn means, no awkward kinks, or pinch points, or abrupt discontinuities in the function or any of its derivatives. But we haven’t defined differentiation on manifolds so to make smooth structures work we need to use collections of sets that cover a manifold and pair the collection with coordinate charts. Then we say a function f is smooth if (and only if) the function composed with the inverse chart is smooth. Fair enough, but in order for this to work we would like to not be bogged down by deficiencies in the chart so to guarantee that we are focusing on the function we develop a structure called an atlas, which is a collection of sets whose domain is the whole manifold and whose charts are “smoothly compatible”, meaning they don’t hit the same points, or compose smoothly with the inverse of the others.

Turns out that a maximal atlas (any other atlas is contained in the maximal) + topological manifold = smooth manifold.

That’s it for chapter 1. I’m going to grab a physics textbook (General relativity) and avail myself of actual coordinate systems and objects I can calculate with. Will post my thoughts on chapter 2, Smooth Maps, in a few days.