Abstract Nonsense

Surfaces (Pt. I)

Point of Post: In this post we start discussing surfaces, giving their motivation, as well as their formal definition.

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Motivation

In this post we finally begin discussing the real meat, the cool part of geometry (at least the narrow part of it I am currently learning). In particular, in this post we start discussing surfaces. I imagine everyone who would read this understands, at least from an intuitive non-mathematically minded sense, what a surface is. This is simultaneously a fantastic and horrible situation. The fact that every average Joe and Janet knows what a surface is tells one that they are, if nothing else, prevalent in the real world, and in particular, the academic sphere. Also, this familiarity with the idea of a surface often gives one intuition about what precisely is happening, or more often then not, what precisely should happen. That said, having an idea about what a surface is means that when the, somewhat involved, definition of a surface is given, people are upset that isn’t as ‘simple’ as they’d hoped. Moreover, this preconceived notions often times leads to false intuition, beliefs that are wildly false.

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Regardless, what is a surface, and why are they important? Surfaces can be, and almost always are, best described as ‘locally $2$-Euclidean’ spaces. Of course, this just transfers the intuitive idea of surface to the seemingly complicated idea of ‘locally $2$-Euclidean’. Luckily though, although scary sounding, being locally $2$-Euclidean is a very concrete and visually appealing idea. Allow me some license, and imagine that one had a gift wrapped Christmas present sitting in front of them right now. If asked “what is the dimensionality of the wrapping paper enveloping the present” one is apt to say “Oh, well, clearly the wrapping paper is a three-dimensional object!” Indeed, the wrapping paper ‘takes up three dimensions’ in the sense that it extends in all three spacial dimensions, etc. But, this was only after the work of actually wrapping the present. For example, suppose that this was a very large present, and it had taken several sheets of wrapping paper, taped together, to form the wrapping paper cover of the present. So, while the entirety of the taped-together wrapping paper cover is a ‘three dimensional object’ it is comprised of several sheets of wrapping paper which are (being just ‘planes’) two-dimensional objects. Another way to say this is that each point on the wrapping paper cover lives on one of the individual wrapping paper sheets, and so to the point, it might as well be living in two-dimensional space since all it ‘sees’ is the two dimensional sheet it inhabits. This is precisely what locally $2$-Euclidean seeks to capture in its definition. Namely, a space is locally $2$-Euclidean if it can be ‘built’ out of $2$-dimensional objects, or more precisely, if around each point the space ‘looks like’ a $2$-dimensional object. The classic example put into the discussion of surfaces is the surface of the Earth. No one would dispute that the surface of the earth is a three-dimensional object, but to each of us (the ‘points’) what we see are flat floors and straight line horizons. In other words, to us the Earth looks flat, it looks like the plane $\mathbb{R}^2$.

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So, why are surfaces important? Besides the obvious answer that surfaces are all around us, they are the spaces on which we operate, on which we live, there are more mathematically interesting. Surfaces encompass two of the most interesting kinds of geometric objects: common loci of sets of functions and graphs. To be more explicit, given a sufficiently nice mapping $g:\mathbb{R}^3\to\mathbb{R}$ and sufficiently nice points $c\in\mathbb{R}$ the level set $\left\{(x,y,z)\in\mathbb{R}^3:g(x,y,z)=c\right\}$ will be surfaces. Moreover, if $f:\mathbb{R}^2\to\mathbb{R}$ is a sufficiently nice function then the graph $\Gamma_f=\left\{(x,y,f(x,y)):x,y\in\mathbb{R}\right\}$ will be a surface. These are two fundamental objects associated to functions $\mathbb{R}^3\to\mathbb{R}$ and $\mathbb{R}^2\to\mathbb{R}$. Indeed, the rigorous notion of a surface has its origins (although certainly not exclusively) upon the considerations of such objects (graphs and loci) when studying the functions themselves.

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Something that was omitted in the above description, hidden implicitly in terms such as ‘sufficiently nice’, is that we for us, for the kind of geometry we want to do, plain old locally $2$-Euclidean spaces aren’t going to be good enough. Indeed, what we’d like to do on surfaces is do calculus. We’d like to define notions of differentiability and integrability, etc. so that we can study the geometric properties of surfaces via analytic methods. Consequently, certain types of surfaces which evidently satisfy the locally $2$-Euclidean (such as the wrapping paper example (depending on precisely what was meant by the wrapping paper)). So, how precisely do we hope to do calculus on surfaces? Well, at least how we plan to do differential calculus should be intuitive enough. Namely, since differential calculus is a local theory we should be able to appeal to the local Euclideaness of our spaces to do calculus much the same as we would do in $\mathbb{R}^n$.

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Topological and Smooth Surfaces

We now make precise what we mean by locally $2$-Euclidean. Namely, we call a subset $\mathcal{S}$ of $\mathbb{R}^3$ a topological surface if for every $p\in\mathcal{S}$ there exists a neighborhood (in $\mathcal{S}$ with the subspace topology) $U$ of $p$ which is homeomorphic to an open subset of $\mathbb{R}^2$. If $\varphi:\varphi^{-1}(U)\to U$ is such a homeomorphism (with $\varphi^{-1}(U)\subseteq\mathbb{R}^2$ open) we call the ordered pair $\left(U,\varphi\right)$ a topological chart at $p$ or just a topological chart when the base point isn’t important. A collection $\mathfrak{A}=\left\{\left(U_\alpha,\varphi_\alpha\right)\right\}_{\alpha\in\mathcal{A}}$ of charts on $\mathcal{S}$  such that $\displaystyle \mathcal{S}=\bigcup_{\alpha\in\mathcal{A}}U_\alpha$ is called an topological atlas for $\mathcal{S}$. We see then that a subset $\mathcal{S}$ of $\mathbb{R}^3$ is a topological surface if and only if it admits an atlas. We define the transition map $\tau_{\alpha,\beta}$ between the intersecting charts $(U_\alpha,\varphi_\alpha)$ and $(U_\beta,\varphi_\beta)$ to be the map $\tau_{\alpha,\beta}:\varphi_{\alpha}^{-1}(U_\alpha\cap U_\beta)\to U_\alpha\cap U_\beta$ given by $\tau_{\alpha,\beta}=\varphi_\beta\circ\varphi_{\alpha}^{-1}$.

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With this in mind, we can now define the objects of our future study. A subset $\mathcal{S}$ of $\mathbb{R}^3$ is called a smooth regular surface or just surface if it admits a topological atlas $\mathfrak{A}$ such that for every $\left(U,\varphi\right)\in\mathfrak{A}$ the map $\varphi:\varphi^{-1}(U)\to U$ is smooth and the total derivative $D_\varphi(p):\mathbb{R}^2\to\mathbb{R}^3$ is injective for each $p\in \varphi^{-1}(U)$. Such a topological atlas is called a smooth regular atlas or just atlas for short.

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We note that since the Jacobian matrix $\text{Jac}_\varphi(p)=\left(\begin{array}{c|c}D_1\varphi(p) & D_2\varphi(p)\end{array}\right)$ (where, as usual if $\varphi(p)=(\varphi_1(p),\varphi_2(p),\varphi_3(p))$ then $D_i(p)=(D_i\varphi_1(p),D_i\varphi_2(p),D_i\varphi_3(p))$ where $D_i$ denotes the $i^{\text{th}}$ partial derivative) that $D_\varphi(p)$ being injective is equivalent to $D_1\varphi(p)$ and $D_2\varphi(p)$ being linearly independent for each $p\in \varphi^{-1}(U)$. To beat a dead horse, we emphasize for computational purposes that $D_\varphi(p)$ being injective is equivalent to $D_1\varphi(p)\times D_2\varphi(p)$ being non-zero, where $\times$ denotes the usual cross product.

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I feel as though at this point some examples are in order. I submit for your approval the absolutely classic example (the abstractification of the surface of the earth example):

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ExampleLet $\mathbb{S}^2$ denote the usual $2$-sphere given by $\left\{\bold{x}\in\mathbb{R}^3:\|\bold{x}\|=1\right\}$. Then, $\mathbb{S}^2$ is a surface. Indeed, for each $p\in\mathbb{S}^2$ we let $U_p=\left\{x\in\mathbb{S}^3:\langle x,p\rangle>0\right\}$. Clearly then $U_p$ is open in $\mathbb{S}^3$. Let $T_p:\mathbb{R}^3\to\mathbb{R}^3$ be some linear isomorphism sending $p\mapsto (0,0,1)$. Consider then the mapping $\varphi_p:\mathbb{D}\to U_p$, where $\mathbb{D}$ is the unit disc in $\mathbb{R}^2$, given by $(x,y)\mapsto T_p(x,y,\sqrt{1-x^2-y^2})$. Then, $\varphi_p$ is easily seen to be a smooth bijection. Noting then that

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$D_{\varphi_p}(x,y)=T_p\circ\begin{pmatrix}1 & 0\\ 0 & 1\\ \displaystyle \frac{-x}{\sqrt{1-x^2-y^2}} & \displaystyle \frac{-y}{\sqrt{1-x^2-y^2}}\end{pmatrix}$

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It’s easy to see that the columns of the inner matrix are linearly independent and so $D_{\varphi_p}(x,y)$ is injective. Thus, we may conclude that $\left\{(U_p,\varphi_p)\right\}_{p\in\mathbb{S}^2}$ is an atlas for $\mathbb{S}^2$ and so $\mathbb{S}^2$ is a surface.

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We submit our second example as more of a theorem:

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Theorem: Let $f:U\to\mathbb{R}$, where $U\subseteq\mathbb{R}^2$ is open, be a smooth function. Then, the graph $\Gamma_f=\left\{(x,y,f(x,y)):(x,y)\in U\right\}$ is a surface.

Proof: Define $\varphi:U\to\Gamma_f:(x,y)\to (x,y,f(x,y))$. Clearly $\varphi$ is a smooth and bijective, with continuous inverse $(x,y,f(x,y))\mapsto (x,y)$. We claim now that $D_\varphi(x,y)$ is injective for each $(x,y)\in U$. To see this we merely note that $D_1\varphi(x,y)=(1,0,D_1f(x,y))$ and $D_2\varphi(x,y)=(0,1,D_2f(x,y))$ and clearly these two are linearly independent, from where it follows that $\varphi$ is a smooth chart. But, since $\varphi(U)=\Gamma_f$ we see that $\left\{(U,\varphi)\right\}$ is an atlas for $\Gamma_f$ from where the conclusion follows. $\blacksquare$

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References:

[1] Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.

[2] Pressley, Andrew. Elementary Differential Geometry. London: Springer, 2001. Print.

[3]  Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.

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October 7, 2011 -

2 Comments »

1. […] Surfaces (Pt. II) Point of Post: This is a continuation of this post. […]

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2. […] our last post we defined the notion of surfaces and gave a few examples, coming to the high point when we showed […]

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