Monday, December 21, 2015

Fall 2015 Sunset, Spring 2016 Dawn

Let me digress for a second: I delete more drafts than I publish. I guess that that is the writing process, as are many other things in life.

Last semester, I lightheartedly complained that algebra was my toughest class. Now that the semester has come to an end, in hindsight, all of my classes were extraordinarily fundamental. There certainly was a lot to take in, but to the advantage of the student, it was basic material; not much wit was necessary to answer the countless questions that were asked, but a solid knowledge and understanding of the definitions and results was.

As I enrolled in my next batch of classes for the coming semester and skimmed through the material that we will be covering, I decided that the most sensible thing that I can do right now is begin studying. I will be taking three courses, which I will arrange in increasing order of difficulty.
  1. Introduction to Real Analysis II
  2. Modern Algebra II
  3. Geometry of Manifolds
I am expecting analysis to be a smooth continuation of the not-too-difficult course that I took last semester. I felt tremendously confident with it and am expecting to feel the same way this time. Algebra will always be algebra: Tough. But since most of the fundamental definitions are already out of the way, I am expecting it to feel not as brutal as it felt last semester—Galois theory aside, of course (that will be insane). Now, geometry of manifolds will be an incarnate demon. It is the natural continuation of topology, a class in which I excelled, but the material is so alien to me right now and the professor so sophisticated that I am decidedly expecting it to be my toughest course by far, ever.

Until classes begin, I will dedicate the next few blog entries to geometry of manifolds, to accrue some basic definitions and results.

Geometry of Manifolds


A topological space $M$ is called a topological $n$-manifold if it has the following properties:
  • $M$ is Hausdorff,
  • $M$ is second countable, and
  • $M$ is locally Euclidean of dimension $n$.
Lemma. If $M$ is a second countable topological space, then every open cover of $M$ has a countable subcover.

A chart on $M$ is a pair $\left(U,\varphi\right)$, where $U$ is an open subset of $M$ and $\varphi:U\to\tilde U$ is a homeomorphism from $U$ to an open subset $\tilde U=\varphi\left(U\right)\subseteq\mathbb R^n$.

If $\left(U,\varphi\right)$ and $\left(V,\psi\right)$ are two charts such that $U\cap V\neq\varnothing$, then the composite map $\psi\circ\varphi^{-1}:\varphi\left(U\cap V\right)\to\psi\left(U\cap V\right)$, called the transition map from $\varphi$ to $\psi$, is a composition of homeomorphisms, and is therefore itself a homeomorphism. Two charts $\left(U,\varphi\right)$ and $\left(V,\psi\right)$ are said to be smoothly compatible if either $U\cap V=\varnothing$ or the transition map $\psi\circ\varphi^{-1}$ is a diffeomorphism.

We define an atlas for $M$ to be a collection of charts whose domains cover $M$. An atlas $\mathcal A$ is called a smooth atlas if any two charts in $\mathcal A$ are smoothly compatible with each other. A smooth atlas $\mathcal A$ on $M$ is called maximal if it is not contained in any strictly larger smooth atlas.

A smooth structure on a topological $n$-manifold $M$ is a maximal smooth atlas. A smooth manifold is a pair $\left(M,\mathcal A\right)$, where $M$ is a topological manifold and $\mathcal A$ is a smooth structure on $M$. Smooth structures are also called differentiable structures or $C^\infty$ structures by some authors.

Lemma. If $M$ is a topological manifold, then every smooth atlas for $M$ is contained in a unique maximal smooth atlas, and two smooth atlases for $M$ determine the same maximal smooth atlas if and only if their union is a smooth atlas.

I need to further peruse and dissect my book and the literature to continue...