Abstract Nonsense

Crushing one theorem at a time

Instructive Non-Examples


Point of Post: In this post we give some instructive non-examples of surfaces including pointy sets (sets with a singularity), non-two dimensional sets, and ‘thick’ subsets of \mathbb{R}^3.

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October 14, 2011 Posted by | Differential Geometry | , , , , , , , | Leave a comment

Surfaces (Pt. III)


Point of Post: This is a continuation of this post.

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October 9, 2011 Posted by | Differential Geometry | , , , , , , , , , | 1 Comment

Surfaces (Pt. II)


Point of Post: This is a continuation of this post.

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October 9, 2011 Posted by | Differential Geometry | , , , , , , , | 2 Comments

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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October 7, 2011 Posted by | Differential Geometry | , , , , , , , , | 2 Comments

Space Curves


Point of Post: In this post we extend our notions of curvature and similar matters from plane curves to space curves.

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Motivation

Although we have been, up until this point, discussing properties of plane curves, the majority of our focus shall be on curves (and surfaces) living in space (i.e. in \mathbb{R}^3).  One would hope that the theory of such curves, in particular the theory of their curvature, is as nice as it was for plane curves. Unfortunately, this is not the case. As we shall see, we can define curvature for a curve in \mathbb{R}^3 very analogously to how we defined it for curves in \mathbb{R}^2, but what we shall see is that unlike the \mathbb{R}^2 case, curvature isn’t the only important invariant (in the sense that a space curve is not, except for an isometry, determined by its curvature). Indeed, we will see that the embedded circle \left\{(x,y,0):x^2+y^2\right\} and the helix \left\{(\frac{1}{2}\cos(t),\frac{1}{2}\sin(t),\frac{1}{2}t):t\in\mathbb{R}\right\} have the same curvature functions (constant functions, equal to 1) yet are clearly not differing by an isometry. Clearly the problem is that curvature doesn’t account for ‘upward or downward’ motion and consequently isn’t sensitive enough to differentiate between curves in \mathbb{R}^3. We are thus necessitated to create another statistic for curves which, intuitively at least, measures the up and down movement. This statistic is known as the torsion of the curve and measures what the previous sentence (the intuition of up and down movement) wishes–the extent to which the curve does not stay within a given plane. We shall then see that these two statistics completely determine, as always up to an isometry, space curves–the torsion controlling the movement in and out of planes, and the curvature controlling the movement within each plane.

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October 5, 2011 Posted by | Differential Geometry | , , , , , | 1 Comment

The Fundamental Theorem of Plane Curve Curvature


Point of Post: In this post we prove the fundamental theorem of plane curve curvature and use it to prove the curvature characterization of  segments of circles (positive constant curvature). Moreover, we also show that the fundamental theorem implies the existence of an osculating circle.

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Motivation

We now prove one of the most fulfilling theorems in the differential geometry of \mathbb{R}^2, namely that a smooth unit speed curve is entirely determined up to an orientation preserving isometry of \mathbb{R}^2 by its curvature. This is intuitively obvious for if one draws a curve and decides that it will follow some set ‘curvature pattern’ which is unit speed, then there is only two decisions to make: where to start, and what direction to start towards. But, this is precisely the statement in the first sentence. In fact, we’ll prove more. Not only does the curvature function characterize, up to an isometry, a unit speed curve but in fact given any smooth map f:I\to\mathbb{R} for some interval I\subseteq\mathbb{R} one has that there exists a unit speed curve \mu:(a,b)\to\mathbb{R}^2 such that \varkappa_{\mu}=f. As a particular consequence of  this deep fact is the somewhat satisfying fact that the only curve with positive constant curvature is a segment of a curve. We will then talk about what shall serve to be a prime motivator for curvature in higher dimensions–the osculating circle. Intuitively, given a point on a regular curve there is a unique circle which ‘best approximates’ the curve near that point, this circle is precisely the circle tangent to the curve at that point with radius equal to the inverse of the curvature at that point.

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September 29, 2011 Posted by | Differential Geometry | , , , , , , | 1 Comment

More Geometric Way of Viewing the Curvature of a Plane Curve


Point of Post: In this post we discuss a nicer, more geometric way of thinking about the curvature of a plane curve via its turning angle.

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Motivation

In our last post we discussed a way of measuring the (unsigned) curvature \kappa_\gamma of a curve \gamma. The definition though, while simple and easy to motivate, lacked the geometric flair that, doing geometry, we’d hope to have. So, we use one of the beautiful aspects of differential geometry, the go-between of simple to define and proof analysis notions and the beautiful and intuitive geometry ones. For those who have taken algebraic topology this could be analogized to the relationship between singular and simplicial homology, where the latter’s existence is almost solely to prove the first makes sense.  Regardless, we’d now like t0 formulate a different way of defining curvature which although equivalent to the way defined before (except now we make a sign convention thus arriving at the ‘signed’ curvature) has a much more geometric feel.  To see what this definition is we note, using the formulation from last time, that it suffices to give a reinterpretation for unit speed curves. So, what is this super geometric way of viewing the curvature? Well, since \gamma:I\to\mathbb{R}^2 is unit speed we know that \gamma'(t) is a unit vector, and so is determined uniquely by it’s angle with the positive x-axis. Thus, there exists a function \varphi:I\to\mathbb{R} such that \gamma'(t)=(\cos(\varphi(t)),\sin(\varphi(t)) for all t\in I. That said, it’s obvious that there exists lots of such functions, namely \varphi(t)+2\pi is another such function. The non-obvious, but very important fact is that there exists a unique smooth \varphi:I\to\mathbb{R} which works. This function is what we shall defined as the turning angle. It’s then inuitively obvious that the curvature of \gamma should be the rate of change of the turning angle. This, is precisely what we shall prove. This formulation and the associated theorems will have some very interesting consequences that we shall discuss in our next post.

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September 25, 2011 Posted by | Differential Geometry | , , , , , | 1 Comment

Curvature of Plane Curves


Point of Post: In this post we motivate the idea of the curvature of a plane curve in preparation for higher dimensional notions of curvature. We then derive the general formula to calculate the curvature.

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Motivation

We come now to what could be easily stated as the focus of the geometry of curves and surfaces (at least at this level): curvature. I have faith that anyone reading this has at least an intuitive notion of what ‘curvature’ is. It’s how much the curve ‘twists’, or ‘moves around’, or any other equally synonymous term. As always though, we need to find a way to transfer our vague intuitions into hard mathematics. So, let’s see if we can find some way of mathematically describing this ‘twisting’. Well, whatever we may define curvature to be, I doubt that anyone would argue that the curvature of a line should be zero. Thus, the curvature of a curve should mention the deviation that curve takes from a straight line–the question is, which straight line? To make the next leap a little more obvious, let’s take a second and try to relate what we’re doing to a real world scenario. Perhaps where curvature affects us most is when we are driving. No one likes when they accidentally come up to a curve too fast and have difficulty making it. So, what exact property made it hard to make? Well, it’s not hard to see that the difficulty is that we are initially going in a certain direction, and it’s the turning away from this direction (that we are going at some instantaneous point) that’s hard. So, the line we seek to measure deviation from is the ‘trajectory line’, whatever that means. But, a seconds thought gives us that the trajectory line is the line passing through our current position in the direction of our velocity. Aha! That’s it, since velocity is the derivative of position, it’s not hard to take this analysis and see that our ‘trajectory line’ is nothing more than the line spanned by the tangent vector at our current position. Thus, the curvature of a curve \gamma at a point \gamma(t_0) should be the quickeness at which \gamma deviates from \gamma'(t_0). So, how exactly do we do this? There are precisely two unit vectors n,-n perpendicular to \gamma'(t_0), pick one. Since they differ by a sign it’s not important (at this point) which we pick, so let’s say n. We then want to measure, for small \Delta_t, the change from d(\ell,\gamma(t_0)) to d(\ell,\gamma(t_0+\Delta_t)) where \ell denotes \text{span}(n) and d denotes the distance between a point and a set. But, since n is normal it’s not hard to see that this difference is equal to (\gamma(t_0+\Delta_t)-\gamma(t_0))\cdot n. But, expanding \gamma(t_0+\Delta_t) as a Taylor series we see that it’s equal to

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\gamma(t_0+\Delta_t)=\gamma(t_0)+\gamma'(t_0)\Delta_t+\frac{1}{2}\gamma''(t_0)\Delta_t^2+O(\Delta_t^3)

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and so (\gamma(t_0+\Delta_t)-\gamma(t_0))\cdot n is equal to

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\Delta_t\gamma(t_0)\cdot n+\frac{1}{2}\Delta_t^2\gamma''(t_0)\cdot n+O(\Delta_t^3)

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Now, since n is perpendicular to \gamma'(t_0) by construction we see that the change in distances is equal to \frac{1}{2}\Delta_t^2\gamma''(t_0)\cdot n+O(\Delta_t^3). Now, if we restrict ourselves to unit speed curves we shall prove that \gamma'(t_0) is perpendicular to \gamma''(t_0) and so consequently \gamma''(t_0) is parallel to n. Thus, we may finally conclude that the magnitude of the difference of the distances is \frac{1}{2}\|\gamma''(t_0)\|\Delta_t^2+O(\Delta_t^3). So, forgetting the factor of \frac{1}{2} (we are trying to define a consistent statistic for curves, multiplying by a constant is irrelevant) we see that the only important number as \Delta_t\to 0 is \|\gamma''(t_0)\|. Ta-da! Thus, the (unsigned) curvature of a unit speed curve \gamma at \gamma(t_0) is just \|\gamma''(t_0)\|. Thus, if \|\gamma''(t_0)\| is large then the curve curves ‘a lot’ and conversely. But, you’ll notice that we’ve restricted ourselves to unit speed curves in our definition. But, this is an easy fix. Indeed, as indicated in our last post the only curves we shall really concern ourselves will be regular curves, and we know that every regular curve has a unit speed reparamaterization from where it’s pretty clear that we should just define the curvature of the regular curve to be equal to the curvature of any of its unit speed reparamaterizations. Ostensibly, there may be an issue here, in the sense of ‘which unit speed reparamaterization do we pick’. But, the problem isn’t a problem at all since all unit speed reparamaterizations of a given regular curve will have the same curvature. This make sense intuitively since the curvature of a curve is an intrinsically geometric idea which should be independent of paramaterization. So, with all this in mind, let’s do some math.

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September 24, 2011 Posted by | Differential Geometry | , , , , | 3 Comments

Closed Curves


Point of Post: In this post we discuss the notion of closed curves, proving such facts such as the existence of a minimal period.

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Motivation

Most times when people think of honest to god curves they often think of curves that ‘closed up’ in the sense that eventually the curve curves back over itself. A prime example of this is the classic unit circle \mathbb{S}^1. It’s not hard to see that such plane curves are described parametrically by periodic functions of the input variable t. One may then begin to ask, is there a ‘fundamental period’ for such a parametrized curve? By this, I mean that for a given curve with one number T with the property that \gamma(t+T)=\gamma(t) for all t\in I then clearly the number 2T has the same property. In particular, nT for every n\in\mathbb{Z} also has this property. One may then begin to ask whether there is a ‘smallest’ positive such period. The answer turns out to be yes for a very cool reason.

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September 23, 2011 Posted by | Differential Geometry, Topological Groups | , , , , , , | Leave a comment

Reparamaterization, Regular Curves, and Unit Speed Curves (Pt. I)


Point of Post: In this post we discuss the notion of reparamaterizations (definitions and intuition), regular curves, unit curves, and the interrelation between the three.

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Motivation

Up to this point we’ve figured out that intuitively the better way to discuss a ‘curve’ thought of as a subset of some Euclidean space is to instead imagine a smooth mapping \gamma:I\to\mathbb{R}^n whose image is equal to the ‘curve’ we are focusing on. That said, an obvious question that comes with the territory is–which map do we choose? In other words, multiple maps can paramaterize the same ‘curve’. For example, it’s not hard to see that

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\begin{aligned}&\gamma:\mathbb{R}\to\mathbb{R}^2:t\mapsto (\cos(t),\sin(t))\\ &\alpha:\mathbb{R}\to\mathbb{R}^2:t\mapsto (\cos(2t),\sin(2t))\\ &\beta:\mathbb{R}\to\mathbb{R}^2:t\mapsto (\sin(t),\cos(t))\end{aligned}

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all have the unit circle \mathbb{S}^1 as their image. So, if these smooth maps are our vehicle to study ‘curves’, then clearly whatever properties we care about should be the same for whichever smooth curve we pick to represent the ‘curve’. But, it’s clear that all not curves are created equal. For example, I think it’s pretty clear that the functions \mathbb{R}^+\to\mathbb{R}^2 given by t\mapsto (t,t^2) and t\mapsto (t^2,t^4) have the same image, but are fundamentally different. In particular, if one looks at the tangent vectors to each of these curves at t=0 one finds that the first has non-zero tangent vector but the second doesn’t. So? Why does this matter? Well, it shall soon be important to us that we are able to make statements such as “there is precisely two unit vectors orthogonal to \gamma'(t)” which is, as I’m sure you are well aware, is always true–except when \gamma'(t)=0. Thus, as we shall see it will be a decidedly geometrically important fact as to whether or not a curve has a zero tangent vector. In particular, two curves one of which has this property and one of which don’t are fundamentally different objects. So, our notion of equivalence between curves, should be discerning enough to detect this, so that for, example (t,t^2) and (t^2,t^4) are not equivalent but (perhaps) (\cos(t),\sin(t)) and (\sin(t),\cos(t)) are. Moreover, we shall see that curves that are ‘nice’ (in the sense that they never have a zero tangent vector) are equivalent (whatever this will eventually mean) to the nicest possible type of curve, namely a curve whose tangent vectors are always unit vectors. This type of curve is nice for both now-hidden and now-apparent reasons. For example, for such a curve it’s pretty easy to calculate arc length since \|\gamma'(t)\|=1.

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September 23, 2011 Posted by | Differential Geometry | , , , , , , , , | 5 Comments

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