guide:91a4bd3292

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Back now to applied mathematics, involving analysis and physics, the situation here is a bit different. Although many interesting algebraic manifolds appear at the advanced level, making algebraic geometry a key tool in advanced physics, in what concerns the basics, here we are mostly in need of a different definition, as follows:

Definition

A smooth manifold is a space [math]X[/math] which is locally isomorphic to [math]\mathbb R^N[/math]. To be more precise, this space [math]X[/math] must be covered by charts, bijectively mapping open pieces of it to open pieces of [math]\mathbb R^N[/math], with the changes of charts being [math]C^\infty[/math] functions.

As a basic example, we have [math]\mathbb R^N[/math] itself, or any open subset [math]X\subset\mathbb R^N[/math]. Another example is the circle, or curves like ellipses and so on, for obvious reasons. To be more precise, the unit circle can be covered by 2 charts as above, by using polar coordinates, in the obvious way, and then by applying dilations, translations and other such transformations, namely bijections which are smooth, we obtain a whole menagery of circle-looking manifolds.

In particular, we can see from this that Definition 11.22 is a serious rival to Definition 11.11, because both generalize, in a natural way, the conics that we know well.

So, this is the situation, we have two geometric disciplines inspired by the findings of the old Greeks, regarding the conics, called algebraic geometry and differential geometry. With algebraic geometry being substantially older, but in what concerns us, we will rather go for differential geometry, which is more relevant to basic modern physics.

Going back now to Definition 11.22 as stated, let us first explore the basic examples. We have already talked about them in the above, but all that discussion needs to be accompanied by some proofs, and here is a more precise statement in this sense:

Theorem

The following are smooth manifolds, in the plane:

• The circles.
• The ellipses.
• The non-degenerate conics.
• Smooth deformations of these.

Show Proof

All this is quite intuitive, the idea being as follows:

(1) Consider the unit circle, [math]x^2+y^2=1[/math]. We can write then [math]x=\cos t[/math], [math]y=\sin t[/math], with [math]t\in[0,2\pi)[/math], and we seem to have here the solution to our problem, just using 1 chart. But this is of course wrong, because [math][0,2\pi)[/math] is not open, and we have a problem at [math]0[/math]. In practice we need to use 2 such charts, say with the first one being with [math]t\in(0,3\pi/2)[/math], and the second one being with [math]t\in(\pi,5\pi/2)[/math]. As for the fact that the change of charts is indeed smooth, this comes by writing down the formulae, or just thinking a bit, and arguing that this change of chart being actually a translation, it is automatically linear.

(2) This follows from (1), by pulling the circle in both the [math]Ox[/math] and [math]Oy[/math] directions, and the formulae here, based on Theorem 11.6 or Theorem 11.7, are left to you, reader.

(3) We already have the ellipses, and the case of the parabolas and hyperbolas is elementary as well, and in fact simpler than the case of the ellipses. Indeed, a parablola is clearly homeomorphic to [math]\mathbb R[/math], and a hyperbola, to two copies of [math]\mathbb R[/math].

(4) This is something which is clear too, depending of course on what exactly we mean by “smooth deformation”, and by using a bit of multivariable calculus if needed.

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In higher dimensions now, as basic examples here, we have the unit sphere in [math]\mathbb R^N[/math], and smooth deformations of it, once again, somehow by obvious reasons. In case you are wondering on how to construct explicit charts for the sphere, the answer comes from:

Theorem

We have spherical coordinates in [math]N[/math] dimensions,

[[math]] \begin{cases} x_1\!\!\!&=\ r\cos t_1\\ x_2\!\!\!&=\ r\sin t_1\cos t_2\\ \vdots\\ x_{N-1}\!\!\!&=\ r\sin t_1\sin t_2\ldots\sin t_{N-2}\cos t_{N-1}\\ x_N\!\!\!&=\ r\sin t_1\sin t_2\ldots\sin t_{N-2}\sin t_{N-1} \end{cases} [[/math]]

with this guaranteeing that the sphere is indeed a smooth manifold.

Show Proof

There are several things going on here, the idea being as follows:

(1) The fact that we have indeed spherical coordinates is clear, with the only point to be clarified being the identification of the precise ranges of the angles, which follows from some geometric thinking, first at [math]N=2,3[/math], and then in general.

(2) With the remark that we use here mathematicians' convention for the angles, which works nicely in [math]N[/math] arbitrary dimensions, as opposed to physicists' convention, which works best at [math]N=3[/math]. And with this being something quite subjective, because mathematicians' convention is based on physicists' finding that the more dimensions, the better.

(3) Finally, in what regards the last assertion, this can proved a bit like for the circle, as we did in the proof of Theorem 11.23 (1), basically by cutting the sphere into [math]2^N[/math] parts, and we will leave the details here as an instructive exercise.

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Summarizing, we have the spheres as main examples of differential manifolds, as expected, but we can also see from the above that the technical verification of the manifold axioms is something quite complicated. Welcome to differential geometry, where nothing is really trivial, I mean even things which are supposed to be trivial are not.

In relation with these questions, namely parametrizing the spheres, we have the stereographic projection as well, which works more directly, as follows:

Theorem

The stereographic projection is given by inverse maps

[[math]] \Phi:\mathbb R^N\to S^N_\mathbb R-\{\infty\}\quad,\quad \Psi:S^N_\mathbb R-\{\infty\}\to\mathbb R^N [[/math]]

given by the following formulae,

[[math]] \Phi(v)=(1,0)+\frac{2}{1+||v||^2}\,(-1,v)\quad,\quad \Psi(c,x)=\frac{x}{1-c} [[/math]]

with the convention [math]\mathbb R^{N+1}=\mathbb R\times\mathbb R^N[/math], and with the coordinate of [math]\mathbb R[/math] denoted [math]x_0[/math], and with the coordinates of [math]\mathbb R^N[/math] denoted [math]x_1,\ldots,x_N[/math].

Show Proof

We are looking for the formulae of the isomorphism [math]\mathbb R^N\simeq S^N_\mathbb R-\{\infty\}[/math], obtained by identifying [math]\mathbb R^N=\mathbb R^N\times\{0\}\subset\mathbb R^{N+1}[/math] with the unit sphere [math]S^N_\mathbb R\subset\mathbb R^{N+1}[/math], with the convention that the point which is added is [math]\infty=(1,0,\ldots,0)[/math], via the stereographic projection. That is, we need the precise formulae of two inverse maps, as follows:

[[math]] \Phi:\mathbb R^N\to S^N_\mathbb R-\{\infty\}\quad,\quad \Psi:S^N_\mathbb R-\{\infty\}\to\mathbb R^N [[/math]]

In one sense, according to our conventions above, we must have a formula as follows for our map [math]\Phi[/math], with the parameter [math]t\in(0,1)[/math] being such that [math]||\Phi(v)||=1[/math]:

[[math]] \Phi(v)=t(0,v)+(1-t)(1,0) [[/math]]

The equation for the parameter [math]t\in(0,1)[/math] can be solved as follows:

[[math]] \begin{eqnarray*} (1-t)^2+t^2||v||^2=1 &\iff&t^2(1+||v||^2)=2t\\ &\iff&t=\frac{2}{1+||v||^2} \end{eqnarray*} [[/math]]

We conclude that the formula of the map [math]\Phi[/math] is as follows:

[[math]] \Phi(v)=(1,0)+\frac{2}{1+||v||^2}\,(-1,v) [[/math]]

In the other sense now we must have, for a certain [math]\alpha\in\mathbb R[/math]:

[[math]] (0,\Psi(c,x))=\alpha(c,x)+(1-\alpha)(1,0) [[/math]]

But from [math]\alpha c+1-\alpha=0[/math] we get the following formula for the parameter [math]\alpha[/math]:

[[math]] \alpha=\frac{1}{1-c} [[/math]]

We conclude that the formula of the map [math]\Psi[/math] is as follows:

[[math]] \Psi(c,x)=\frac{x}{1-c} [[/math]]

Here, as before, we use the convention in the statement, namely [math]\mathbb R^{N+1}=\mathbb R\times\mathbb R^N[/math], with the coordinate of [math]\mathbb R[/math] denoted [math]x_0[/math], and with the coordinates of [math]\mathbb R^N[/math] denoted [math]x_1,\ldots,x_N[/math].

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There are of course many other possible parametrizations of the sphere, such as the one using cylindrical coordinates, or the one peeling the sphere as an orange, and so on. All this is quite interesting, and as question here, of practical interest, we have:

\begin{question} What is the best parametrization of the unit sphere in [math]\mathbb R^3[/math], for purely mathematical reasons? What about for cartography reasons? \end{question} To be more precise, the first question, which is quite challenging, is that of finding the simplest proof ever for the fact that the sphere in [math]\mathbb R^3[/math] is indeed a smooth manifold. As for the second question, which is even more challenging, and not really solved by mankind, despite centuries of work, and many bright ideas, the problem here is to have your charts reflecting as nicely as possible useful things such as lengths, angles and areas.

By the way, speaking lengths, angles and areas, observe that the general differential manifold formalism from Definition 11.22 is obviously too broad for talking about these. In order to do so, several more axioms must be added, and we end up with something called Riemannian manifold, which is the main object of study of advanced differential geometry. And, regarding such manifolds, there are many deep theorems, including a key result of Nash, stating that we can always find an embedding, as follows:

[[math]] X\subset\mathbb R^N [[/math]]

Thus, all in all, all this leads us into the good old [math]\mathbb R^N[/math], and multivariable calculus. But the story here is quite long and technical, and it's getting late, and we will stop here.

So long for geometry, in a large sense. As a conclusion to all this, geometry comes in many flavors, algebraic or differential, Riemannian or not, over [math]\mathbb R[/math], or [math]\mathbb C[/math], or some other field [math]F[/math], and in addition to this we can talk about affine or projective geometry, or about discrete or continuous geometry, and so on. Many interesting things, and if excited by all this, orient yourself towards physics, where geometry in all its flavors is needed.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].