"Immediate" Applications of Differential Geometry

Question

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another example my teacher presented is proving that on a soccer ball which is made of regular pentagons and regular hexagons, the number of pentagons is fixed, as a consequence of Euler's polyhedral formula. I guess that there are many more of these.

The idea is to find things/facts whose explanation is a theorem in differential geometry and eventually give a reference to a book/article where these connections are explained.

My teacher wants us to make such a list and then each to pick a subject and make a project which presents the theorem which is applied (with proof, if the proof is not too long) and then present the application itself.

Any reference or book on the subject is more than welcome.

Answer 1

When wrapping a ball as a birthday or Christmas present, one cannot avoid the need to crease the paper. This is due to the paper having zero Gaussian curvature, and the ball having positive Gaussian curvature. (Theorema Egregium)

Answer 2

At any given time, there is at least one pair of antipodal points on the Earth that have exactly the same temperature and the same atmospheric pressure. This is a consequence of the "hairy ball" theorem.

Answer 3

The roof of the Olympic arena in Munich is designed using minimal surface techniques. Try Googling the architect's name, Frei Otto. Minimal surfaces, or, (taking gravity into account) surfaces of constant mean curvature are the model for soap films and soap bubbles.

Planes tend to fly along geodesic lines. Shortest distance problems in smooth contexts always involve geodesics.

Classical general relativity is all about curvature and geodesics in special (Lorentz) manifolds. I don't know the current literature, but when I used to work in that area, the books of Hawking and Ellis and the book by Beem and Ehrlich were good resources for a mathematician.

Answer 4

You might consider Sylvester's law of inertia as a theorem in differential geometry. It explains the stability of rotations of rigid bodies. The field of control theory is full of applications of differential geometry, for instance many jet aircraft aren't inherently stable.

The Coriolis force has a purely geometric explanation. One implication is that hurricanes very rarely form within a few degrees of the equator. There's also a measurable effect called "beta drift" due to change in Coriolis with latitude.

Can't forget general relativity! It even needs to be taken into account for GPS to be as accurate as it is (due to the time differences which occur when you put super accurate clocks in orbit).

Answer 5

One nice application is to justify the fact that every map of the globe must distort something (areas, lengths, angles, etc.), that is, there are no faithful (flat) maps of the globe.

Answer 6

Differential geometry explains why your telephone cord gets knotted. Most people pick up the telephone receiver with one of their hands- WLOG with the right hand. So when they pick it up and put it down, they make a clockwise motion. This creates writhe. Writhe plus twist gives linking (this is sometimes known as Călugăreanu's Theorem), and the telephone chords gets supercoiled, and can become knotted because linking relaxes to loops, and anything passing through those loops will create trefoil knots.

Indeed, the above process, to the best of my knowledge, outlines how all knotting occurs in nature. Naturally occurring knots have high writhe- lots of trefoils and torus knots, very few Figure Eight Knots.

Answer 7

In proper differential geometric notation, the fundamental equations underlying electrodynamics, namely the Maxwell's equations, take the following form:

$$dF=0,$$ $$d*F=J,$$ where $F$ is the electromagnetic field strength, $J$ is the current (for example electrons) and $*$ is the Hodge star operator. Since the first equation just says that $F\ $ is closed, and since the four dimensional spacetime topology is usually considered to be uncomplicated, by the Poincaré lemma we can introduce a 1-form $$A=\sum_{\mu=0}^3A_{\mu}dx^{\mu},$$ the electromagnetic four-potential, which satisfies $$F=dA.$$

There are many ways to go from here. I just saw that at the bottom of the Poincaré lemma wikipieda page, they elaborate more on the magnetic aspect of this. As you can see, this is the part which deals with the spatial components of $A$, i.e. the bottom three components. Therefore I think it's worth considering the other extreme here: Simple situations where $A$ has actually only one component, i.e. $A=A_0dx^0\equiv\phi\ dt.$ This is what physicists like to call "the theory of electrostatics". (I'm allowed to talk like this, I'm a physicist myself.) The line that follows $$F=dA=d(\phi\ dt)=d\phi\wedge dt=-dt \wedge d\phi=dt \wedge \left(-\sum_{\mu=0}^3\frac{\partial\phi}{\partial x^{\mu}}dx^{\mu}\right)\equiv dt\wedge E,$$ might be recognisable. It's the cotangent version of $\vec{E}=-\ grad(\phi).$ How this $\vec{E}$ is related to the exact 1-form $E$, whos components are just the time components of the two form $F$ , is determined by the spacetime metric. The components are identical for in flat spacetime. A remark regarding $E$ or $d(-\phi)$ in this context: You can see that, naturally, here the Poincaré lemma hits again - this time in three-dimensional space, where the Maxwell equations state that $curl(\vec{E})=\vec{0}$. So if you don't consider the full theory and exclude time dependence, then you can start from this relation.

Now define the electrostatic force $\vec{F}:=Q\ \vec{E}$, where $Q$ is an electric charge, together with the usual definition of work $W:=\int \vec{F}\ d\vec{s}$ as well as Stokes theorem (or really just the fundamental theorem of calculus), and your everyday school electrostatics follows at once: $$W=\int \vec{F}\ d\vec{s}=\int Q\ \vec{E}\ d\vec{s}=-Q\int grad(\phi)\ d\vec{s}=-Q\ (\phi_2-\phi_1)\equiv Q\ U,$$ where the potential difference $U$ is called voltage.

Another interesting point is the introduction of gauges, which reflect that due to $d^2=0$ we find that $F=d(A+d\alpha)$ holds for any $\alpha$. There are several $A$ for only one electromagnetic field $F$. Broad generalizations of this concepts eventually lead to all the other physical theories involving charges, like the theory of quarks. In fact the whole standard model of particle physics, which describes three of the four fundamental forces, takes this route. This then covers basically all of physics, except for gravity. But, oh snap, that's differential geometry as well!

(On that note, all the differential geometric aspects of the beautiful manifolds termed Lie groups would make an equally long post and another good application. Rotation symmetry, forced on us by the spacetime metric model, the related rotation group $SO(3)$, or more specifically it's universal cover $SU(2)$, and its representations ...they introduce the notion of spin, which eventually explains the stability of matter.)

Answer 8

This one is especially appropriate for a class project: It's well known that if you're eating a slice of pizza and the front end sags, you're supposed to fold it lengthwise (so that the crust gets folded against itself). The reason for this is much less well known: pizza has Gaussian curvature 0, so if you create curvature in the left-right direction, it will be forced to remain straight in the lengthwise direction!

(I'm pretty sure I stole this from an old MathOverflow question.)

Answer 9

If you speak German then this is a nice application of Stokes' theorem to build a planimeter - a device with which you follow an arbitrary curve on paper, and it will determine the area it encloses.

Answer 10

The book "Geometric Control of Mechanical Systems" by Bullo and Lewis is all about using differential geometry for control systems. You can see the corresponding lecture notes here.