Part 3 - Geometry I

August 2019

Part 2 - Information Theory Part 4

"The geometry of innocent flesh on the bone/
causes Galileo's math book to get thrown/
at Delilah, who's sitting, wothlessly alone/
but the tears on her cheeks are from laughter"
- Bob Dylan, "Tombstone Blues"

Index

Introduction
Elements of Topology
The Topological Manifold

Examples 1

3.1 - Introduction

If you have made it this far, I am appreciative of your persistance, but I warn you that the following is going to get into some deeper mathematical ideas. I am not going to water down the technicality in this section, though I will try my best to explain the concepts well enough that I won't have to. If you would prefer more expository view of concepts in geometry without the mathematics I have no suggestions. Surely such things exist, it is simply that I trust the reader's conversance with a search engine is as strong as mine.

I am writing this series of notes for two reasons: first, to consolidate some of my own knowledge about black hole thermodynamics and its priors, and secondly as a primer for more developed notes and future lecture material. With these goals in mind, I will elaborate on the approach taken. I am choosing to write a substantial amount on geometry and its prerequisites in this section and the next because my own approach towards black hole thermodynamics has been deeply mathematical. I consider a strong grounding in differential geometry to be critically important for understanding this deeply theoretical field.

There are portions of mathematics that I will assume the reader is familiar with. Explicitly, I will assume an understanding of multivariable calculus (including vector calculus), linear algebra, and both ordinary and partial differential equations. I will also assume later (though not in this section) a basic understanding of probability and statistical theory. In these topics I will trust the reader to follow along in calculations without much guidance.

I will be taking for granted that the reader is capable of learning some basic theory without my assistance. There will be portions of these notes where I name a theorem used but do not elaborate on its development — there is only so far I can go down the rabbit hole. The branches of mathematics for which this conceit is relevant will be complex analysis and group theory, as well as parts of topology. We will, however, be making some explicit construction of topology and topological ideas for our grounding in geometry. We begin with this in short order.

3.2 - Elements of Topology

In order to understand topological manifolds, we must understand the structure of topological space. We take the simpler route through open sets rather than the historic neighbourhood approach. The name "open sets" is slightly misleasing and is a historical artifact due to the development of topology as a means towards analytical ends. What we call open sets, in a strict topological setting, are simply elements contained in topological space. It is upon us now to define that space.

DEFINITION 3.2.1: A Topological Space $(X,\mathcal{T})$ is a set $X$ and a collection of subsets of $X$, $\mathcal{T}$ which satisfy the following three properties:

The union of an arbitrary collection of subsets, each members of $\mathcal{T}$, is also open and thus a member of $\mathcal{T}$. This may be written as $$ \text{if } O_{\alpha} \in \mathcal{T} \forall \ \alpha, \ \bigcup_{\alpha} O_{\alpha} \in \mathcal{T}. $$
The finite intersection of elements of $\mathcal{T}$ is an element of $\mathcal{T}$. That is, $$ \text{for } O_1 \ldots O_n \in \mathcal{T},\ \bigcap_{i=1}^n O_i \in \mathcal{T}. $$
The set $X$ and the empty set $\varnothing$ are elements of $\mathcal{T}$.

We require a few more notions before moving on to defining topological manifolds. We desire of those objects the following topological properties: that they are Hausdorff and Second-Countable, and a further non-topolological property discussed later. For now, we state the definitions:

Definition 3.2.2: we call a topological space Hausdorff if for any distinct elements $p,q\ \in X$, it is possible to find open sets $O_p$, $O_q$ $\in \mathcal{T}$ such that $p \in O_p$, $q \in O_q$, $O_p \cap O_q = \varnothing$.

Definition 3.2.3: we call a topological space Second-Countable if there is a countable collection of open sets in $X$ such that any open set in $X$ can be expressed as a union of elements in this collection.

The following definition is not necessary for the description of smooth manifolds in themselves, but does fit here as topological preliminary.

Definition 3.2.4: Say $X$ and $Y$ are topological spaces. We say that a transformation $\phi : X \rightarrow Y$ is a homeomorphism if $\phi$ is injective, surjective, continuous, and has continuos inverse.

In the context of sheer topology, what is meant by "continuous" is that the inverse image of every open set in $Y$ is an open set in $X$. Put plainly, homeomorphisms uniquely map open sets to open sets between topological spaces.

Our last definition before we put these pieces together is as follows:

Definition 3.2.5: we call a topological space locally Euclidean of dimension n if for every element $x \in X$ there exists an open subset of $X$ containing $x$, $U$, such that $U$ is homeomorphic to an open subset of $\mathbb{R}^n$.

3. The Topological Manifold

We use the topological notions above to describe one of the most basic objects in geometry.

Definition 3.3.1: a Topological Manifold of dimension n is a topological space that is Hausdorff, second-countable, and locally n-Euclidean.

This most simple manifold will serve as the basis on which we develop the necessary geometry to study general relativity.