Tensors & Differential Geometry – Understanding Physics and Astronomy

If you recall, there was a post I uploaded some time ago where I talked about manifolds and coordinates as a part of my Basics of Tensor Calculus series. I also noted that the post was incomplete. I now plan to rectify that post by treating manifolds properly in this one. The objective of this post is as follows:

Introduce some basic concepts about topology; in particular, the concept of a homeomorphism is of crucial importance.
Discuss some prerequisite material from multivariable calculus (e.g. diffeomorphisms and smooth functions).
Define what is called an $n$ -dimensional differentiable manifold and provide some remarks on the definitions presented.

Part I. Elementary Concepts in Topology:

As I’ve not talked about what a topology is on this blog, I will try to give a quick idea of what a topology of a set is and from there construct the idea of a topological space from which I define a continuous function between two topological spaces which then leads us to the concept of a homeomorphism.

Therefore, we have the following definition:

Definition. (Topology; Topological Space.) Let $X$ be a set, and let $\tau$ be a collection of subsets of $X$ . Then we say that $\tau$ forms a topology on the set $X$ if the following conditions are met:

$X, \emptyset \in \tau$ ;
$\bigcup_{\alpha\in \mathbb{N}}U_{\alpha}\in \tau$ where $U_{\alpha}$ are open in $X$ .
$\bigcap_{\alpha=1}^{n}U_{\alpha}\in \tau$ .

The coordinate pair $(X,\tau)$ , in which the abscissa is the set and the ordinate is the topology, is referred to as a topological space.

What this definition tells us first is that if we are given a set $X$ , then a subcollection of sets formed from the elements of $X$ is such that the set $X$ itself is contained in this subcollection and that the set that contains no elements from $X$ (that is, the empty set) is also contained in it. Second, it tells us that any union of an arbitrary number of elements from $X$ is contained in the subcollection, and lastly that a finite intersection of elements of $X$ is contained in the subcollection as well. Some treatments will refer to elements of $\tau$ as open sets relative to the set $X$ .

We now make a few further definitions related to topological spaces: that of neighborhoods, Hausdorff spaces, closure, interior, and boundary.

Definition. (Closed Set; Closure; Interior; Boundary.) Let $X$ be a topological space, and let $A$ be a subset of $X$ . Then $A$ is said to be closed if $X\setminus A$ is open in the topology. Additionally, given the subset $A$ we define the closure of $A$ to be the intersection of all closed sets containing $A$ . The interior of $A$ is defined to be the union of all open sets that are contained in $A$ . The boundary of $A$ is defined to be the intersection of the closure of $A$ with the closure of the relative complement $X\setminus A$ . We typically denote each of these as follows: (Closure) $\text{cl}(A)$ ; (Interior) $\text{Int}(A)$ ; and (Boundary) $\partial A$ .

We now define the concepts of neighborhoods and Hausdorff spaces:

Definition. (Neighborhood; Hausdorff Spaces.) Let $X$ be a topological space and let $U$ denote an open set. Then let $x$ be a point in $X$ . Then we say that the open set $U$ which contains the point $x$ is a neighborhood of $x$ . Moreover, the topological space $X$ is called a Hausdorff space if for each pair of points $x, x^{\prime}\in X$ such that $x\neq x^{\prime}$ there exists neighborhoods $U$ and $U^{\prime}$ , of $x$ and $x^{\prime}$ , respectively, for which $U\cap U^{\prime}=\emptyset$ .

Now, let $(X,\tau)$ and $(Y,\tau^{\prime})$ be two topological spaces. We can define a function $\displaystyle f: X\rightarrow Y$ that maps the elements of a topological space $X$ to elements of the topological space $Y$ . We can now define the concept of a homeomorphism:

Definition. (Homeomorphism.) Let $X$ and $Y$ both be topological spaces with topologies $\tau$ and $\tau^{\prime}$ . Also, let $\xi:X\rightarrow Y$ be a continuous bijective mapping. Then we say that $\xi$ is a homeomorphism if the inverse mapping $\xi^{-1}:Y\rightarrow X$ exists and is continuous. In the case for which this holds, we say that the topological spaces $X$ and $Y$ are homeomorphic.

Part II. Smoothness and Diffeomorphisms

Definition. (Smooth Function.) Let $X \subseteq \mathbb{R}^{n}$ be open and assume that $\xi:X\rightarrow \mathbb{R}^{n}$ . We denote the partial derivative of $\xi$ by $\frac{\partial \xi}{\partial x^{\alpha}}(x) \triangleq \frac{\partial^{k}\xi}{\partial x^{i_{k}}}$ , where $\alpha = (i_{1},...,i_{k})$ denotes a multiindex. Thus, for a function $\xi:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is said to be smooth provided its partial derivative exists and is continuous for all $\alpha$ .

Definition. (Diffeomorphism.) Let us consider two open sets $X, Y\subseteq \mathbb{R}^{n}$ and suppose that we have a homeomorphism $\xi:X\rightarrow Y$ . Then $\xi$ is said to be a diffeomorphism between the sets $X$ and $Y$ provided that $\xi$ and its inverse $\xi^{-1}$ are both smooth.

Part III. Smooth Manifolds

We now come to the purpose of the post: the definition of a manifold.

Definition. ( $n$ -Dimensional Differentiable Manifold.) Let $M$ denote a Hausdorff topological space. Then $M$ is said to be an $n$ -dimensional differentiable manifold if for a countable collection of open sets $\{X_{i}\}$ (called coordinate patches) which covers $M$ and for a collection of maps $\{\psi_{i}\}$ (called coordinate maps) the following holds:

For every coordinate map $\psi_{i}:X\rightarrow \mathbb{R}^{n}$ , $\psi_{i}$ defines a homeomorphism to $\mathbb{R}^{n}$ . In other words, $X_{i}$ is homeomorphic to $\mathbb{R}^{n}$ .
Given two overlapping coordinate patches $X_{i}$ and $X_{j}$ with coordinate maps $\psi_{i}$ and $\psi_{j}$ , respectively, said coordinate maps are compatible in the sense that $\psi_{i}(X_{i}\cap X_{j})\rightarrow \psi_{j}(X_{i}\cap X_{j})$ and forms a diffeomorphism.

The two conditions stated above can be difficult to process as presented. To provide a more inuitive way of thinking about it, we note that the first condition, in particular, essentially defines the notion of a manifold being locally Euclidean. Rather, if a coordinate patch (or coordinate neighborhood) is selected on the manifold, then there exists a way to assign Euclidean coordinates to that patch in the usual way. Finally, we have defined manifolds with a notion of differentiability, but it is worthy to note that we can easily define an $n$ -dimensional topological manifold by replacing the diffeomorphism with a standard homeomorphism.

This post took awhile but I’m hoping to continue to post on some of the topics I’ve been researching recently. The next post (whenever I am able to get to it) will likely consider further topics in manifolds and perhaps some elementary homology theory. Until then, clear skies!

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[References used to study these topics: Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists. Paul Renteln. Ch 3. & A Short Course in Differential Topology. Bjorn Ian Dundas. Ch.2 ]

Category: Tensors & Differential Geometry

Manifolds: A Second Glance

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