Visual differential geometry and forms - 1

I am trying something new in this blog. If it works, there will probably be more articles like this. I’m reading a technical book and will blog reading notes from it.

The book I’ll be going through over the next few weeks is Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts by Tristan Needham. A reading group at Google for this book just started. Our first meeting would cover the first act, “The Nature of Space” (that is, the first 2 chapters and the exercises in the third one)¹.

Preface 🔗

Written on Netwonmas of 2019², the preface starts with a quote about the dichotomy of the two main, formal, and ancient pillars of mathematics: geometry and algebra. The quote itself is from a 2002 article, Mathematics in the 20th Century by Sir Michael Atiyah – an article which I recommend reading:

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine. […]when you pass over into algebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning.^{3 4}

Needham starts by saying that books on Differential Geometry paradoxically lack geometry content; instead they are filled with algebra, tensors, indices. The few computer generated pictures don’t illustrate much. This is why this book aims to put the geometry back into the textbook, via 235 hand-drawn diagrams and multiple exercises aimed to build the proper intuition.

The author gives two books as the reason for why he became fascinated with differential geometry: the Gravitation book of Misner, Thorne and Wheeler – a 1,217-pages long tome whose mass gives the book a secondary meaning, a book that I’m planning to read at some point –, and the original Principia of Newton – Philosophiæ Naturalis Principia Mathematica in its full name. Both of these use geometry as the basis of arguments.

Needham introduces the concept of ultimate equality, denoted by \(\asymp\) with the meaning that for two quantities \(A\) and \(B\) depending on a small and vanishing quantity \(\varepsilon\), \(A\) is ultimately equal to \(B\) (\(A \asymp B\)) if in the limit \(\varepsilon \rightarrow 0\) their ratio tends to 1:

\[A \asymp B \Leftrightarrow \lim_{\varepsilon \rightarrow 0}{\frac{A}{B}} = 1\]

If Netwon would have used this notation instead of explicitly writing out ultimately have the ratio of equality, Needham argues, then this would have made Principia easier to understand, and the geometrical methods exposed there would have been in use in more recent developments.⁵

There is one small exercise left for the reader in the middle of the preface: proving that \(\asymp\) is an equivalence relation which also inherits some properties of equality (such as multiplying ultimately equal expressions results in ultimately equal products).

The preface then describes the five parts (acts) of the book, and resolves that algebra is somewhat necessary, but should be done gracefully, only when we must do it. It teases that differential forms – shortened as forms in the future – can simplify various areas of mathematics. For example, all of the theorems of vector calculus turn into a single, much simpler statement under differential forms.

Chapter 1: Euclidean and Non-Euclidean Geometry 🔗

From around 300 BCE until modern times, Euclid’s Elements have been used both as an example of rigorous proof and as an introduction to geometry. Starting from 4 axioms (propositions considered to be trivially true, without requiring a proof, such as given 2 distinct points there is exactly one unique line passing through them – the first axiom), Euclid lays the basis of geometry, first in 2D. At some point, he needs to introduce a 5th axiom, the so called parallel postulate, which in modern form is written as

Given a line \(L\) and a point \(p\) not on the line, there is exactly one line that passes through \(p\) and is parallel to \(L\).

With this, Euclid continues his work on Elements, proves many interesting things about geometry (and number theory too!). But the fact that this axiom was introduced much latter was nagging both him and other mathematicians for nearly 2000 years: what if the parallel postulate can be proven from the other 4?

Around 1830, Nikolai Lobachevsky and Jȧnos Bolyai discovered a geometry where the 5th axiom is false: from a point outside of a line, one could draw at least 2 separate lines which won’t meet the line. They were not the first to discover this, but they were the ones that pushed the discovery to its logical conclusions – even though Bolyai was met with a bad response from Gauss, who has himself discovered some of these facts before but kept them secret and now refused to praise Bolyai for discovering them independently (contrast with Euler who once told Lagrange that he will not publish his results on a new topic until Lagrange publishes his work first).

But there is another way to contradict the fifth axiom: what if there are 0 lines that never met \(L\)? This has been known to be a possibility even from Ancient Greece: the earth is curved and any two meridians on earth will meet at the poles. But, ancient Greeks considered only the extrinsic geometry of the 3D sphere. Plus, the 2D intrinsic geometry on the sphere would also fail the first postulate: there is an infinitude of meridians that pass through the North and South Pole, thus there is an infinity of lines that can be drawn through these 2 points on the spherical geometry.

In any case, it is easily shown that on any triangle on a sphere, the sum of its angles is greater than \(180˚\) (or, in radians, \(\pi\)). This is the subject of a puzzle about a traveler that walks according to cardinal directions, after 3 displacements reaches the starting point and sees a bear; the puzzle then asks for the color of the bear.

If we denote by \(\mathcal{E}\) the excess angle (that is by how much the sum of the 3 triangle angles is greater than \(\pi\)), and by \(\mathcal{A}\) the triangle area then we can prove that on a sphere of radius \(R\):

\[\mathcal{E} = \frac{1}{R^2}\mathcal{A}\]

In fact, there is a similar type of relationship in the hyperbolic geometry of Bolyai-Lobachevsky. So, we can generalize that in any geometry, there is a constant \(\mathcal{K}\) which depends only on the geometry such that

\[\mathcal{E} = \mathcal{K}\mathcal{A}\]

On spherical geometries, \(\mathcal{K}\) is positive, on hyperbolic ones is negative and on the classic Euclidean geometry it is \(0\).

This formula, coupled with the fact that you cannot construct a triangle with negative angles also means that in hyperbolic geometries there is a maximum area that a triangle can have.

It also means that triangles of different sizes in non-euclidean geometries cannot have similar angles. Thus, all the theory of similar triangles from 2D Euclidean geometry cannot be applied outside of that. A deeper conclusion is that we can now define a standard unit of measure for each geometry, something we could not have done in the Euclidean geometry (where the proportionality implied by similar triangles meant that scaling a triangle by the number of centimeters in an inch and replacing a centimeter-marked ruler with an inch-based one would produce the same measurement). For example, we could define this standard unit on spherical geometry to be the length of the equilateral triangle where all angles are at \(90˚\) (one corner at one of the poles and the other 2 on the equator). Of course, a better approach for defining this standard length should take into account \(\mathcal{K}\).

Finally, because of the proportionality with area, on smaller triangles the excess angle tends to 0. Which means that on smaller scales all geometries are Euclidean. Since a human’s size is much much smaller than Earth’s radius, we can perform our lives working with Euclidean geometry for most of the time (until we need long distance travel, etc.)

One more thing discussed in this chapter is the definition of a straight line in each of these geometries, a geodesic: it is the shortest distance between 2 points following the surface. It can be determined by using a string between the two points and wiggling it until it has the shortest possible length, until it is taut. On a squash (or any other vegetable), one could draw such a geodesic, then peel a thin strip containing it, lay this peel on a flat surface and obtain a straight line! Reversing this process gives a very simple method of drawing a line that passes through a point and goes in a certain direction.

Chapter 2: Gaussian curvature 🔗

The Gaussian curvature, or just curvature, is the intrinsic property \(\mathcal{K}\) that we defined above. But, this is not always a constant. Think of the squash from previous section or of a donut (torus in mathematical parlance). In various places on their surface, a tiny triangle can have angles whose sum exceeds \(\pi\) or is less than \(\pi\) in other places. So, curvature is a point property and has to be defined via a limit process: for a point \(p\), consider triangles \(\Delta_p\) around \(p\) that are shrinking to \(p\); then:

\[\mathcal{K(p)} = \lim_{\Delta_p \rightarrow p}{\frac{\mathcal{E}(\Delta_p)}{\mathcal{A}(\Delta_p)}}\]

Although not obvious now, no matter how the triangle shrinks, this limit always exist. On the outside of a donut, it is a positive number (same on a hill, on a sphere, etc.), on the inside it is a negative number (same on Pringles chips, or on a saddle).

There is more to geometry than just triangles, so we can define \(\mathcal{E}\) on n-sided polygons by subtracting more multiples of \(\pi\) for each additional side we add. That is:

\[\mathcal{E}(n{-}gon) = sum\_of\_angles - (n - 2) \pi\]

Unfortunately, this doesn’t help for circles. To solve this, let’s look again at a sphere (of radius \(R\)). Put a pin at the north pole and using a string of length \(r\), draw a circle on the sphere geometry. By computing the difference between the circumference of this circle and the expected \(2\pi r\) and knowing that \(\mathcal{K}\) must be \(\frac{1}{R^2}\), we can extrapolate:

\[\mathcal{K} \asymp \frac{3}{\pi}\left[\frac{2\pi r - C(r)}{r^3}\right]\]

Or, we can use the surface area of this circle and some more math to get:

\[\mathcal{K} \asymp \frac{12}{\pi}\left[\frac{\pi r^2 - \mathcal{A}(r)}{r^4}\right]\]

More interestingly, surfaces with constant \(\mathcal{K} = \frac{1}{R^2}\) are always locally identical to a sphere of radius \(R\) (at some point they might end in a ridge or a sharp point, though). Similarly, surfaces with constant \(\mathcal{K} = -\frac{1}{R^2}\) are identical to the sphere’s equivalent in hyperbolic geometry – the pseudosphere.

Since the excess angle is additive (divide a triangle by a line passing through a corner to prove this) we have the local Gauss-Bonnet theorem that applies on any geometry, even if curvature is not constant:

\[\mathcal{E}(\Delta) = \iint_{\Delta}{\mathcal{K}d\mathcal{A}}\]

That is, get tiny infinitesimal triangles where curvature is almost the same, multiply this with the area of the tiny triangle and sum everything up.

Chapter 3: Exercises for Act 1 🔗

From the prelude, an interesting exercise is to compute

\[\lim_{x \rightarrow 0}{\frac{\sin\tan{x} - \tan\sin{x}}{\sin^{-1}\tan^{-1}{x} - \tan^{-1}\sin^{-1}{x}}}\]

A nice experimental exercise for the first chapter is to take a vase that has a non-constant diameter and launch geodesics in various directions from a point on the circle of maximum diameter. At some point, these geodesics will start curving down! The experiment is to measure this critical angle and determine the relationship between it and the largest and smallest vase diameters. Similar experiments are proposed for the second chapter where the curvature formulas derived above for circles are used to estimate radius of a watermelon and of a pseudosphere (as well as predicting the shape of geodesics on these and on a paper cone).

Good luck solving the limit above!