Harmonicity and Minimalism

Hello, everyone; I hope all is well. In the previous chapter, we discussed the fundamental forms of a surface. There are actually more than two fundamental forms, and we will address that later; for now, let us be content with two. It is clear what we can do with the First Fundamental Form: we are essentially talking about a metric on a surface. In other words, the shortest distance walked by an ant... oh, a geodesic. All right, no more spoilers. Let us begin with a couple of pleasant examples.

Example

Consider the surface $x(u,v) = (u \cos v, u \sin v, u)$. We find its First Fundamental Form to be

$$I = 2du^2 + u^2dv^2$$

As you can see, $F = 0$. In other words,

$$\langle x_u, x_v \rangle = 0$$

It follows that the parametrization is orthogonal. This raises a question: does every surface admit an orthogonal parametrization? The answer is yes, but first we need a definition or two. Before that, let me give you the Second Fundamental Form of this surface—or rather, let us have a look at it. Its normal is calculated as

$$N = (-\cos v, \sin v, 1)$$

The coefficients of the Second Fundamental Form can therefore be calculated from this. There you are: homework.

Definition

If a $(u,v)$ parametrization of a surface $x$ has the property

$$\langle x_u, x_v \rangle = 0$$

then it is called an orthogonal parametrization.

Theorem

Every regular surface locally admits an orthogonal parametrization.

Proof

Ah, would it have been so terrible if I had said we would prove this theorem later, moved on to manifolds, and then pulled out the Weyl tensor or something? Though in that case, what would distinguish me from a university professor? We will need to give a few definitions and deal with complex numbers. But first, let us see what ingredients we have. We return to a topic I also discussed in my master's thesis: recall these operators, our tickets from the real world to the complex one.

Definition (Complex Derivative Operators).

Let our coordinates in $\mathbb{R}^2$ be $(u,v)$. When we define the complex variable $z = u + iv$, the derivative operators change form as follows:

$$\frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial u} - i \frac{\partial}{\partial v} \right), \qquad \frac{\partial}{\partial \bar{z}} = \frac{1}{2} \left( \frac{\partial}{\partial u} + i \frac{\partial}{\partial v} \right)$$

Now take the celebrated First Fundamental Form on our surface,

$$ds^2 = E du^2 + 2F du dv + G dv^2$$

and translate it into the language of these complex operators. With a little algebraic gymnastics—and I am skipping the tedious swarm of intermediate calculations; this is not a mathematics dump, after all—we can write the metric in the elegant form

$$ds^2 = \lambda(z)|dz + \mu(z)d\bar{z}|^2$$

Here, $\lambda$ is a positive real-valued function, while $\mu$ is complex-valued. This is where the rubber meets the road! We want to find a new coordinate system $w$ in which the metric is proportional only to $|dw|^2$, so terms such as $d\bar{w}$ become “waste” and everything is conformal.

For that to happen, the function $w(z)$ must satisfy

$$\frac{\partial w}{\partial \bar{z}} = \mu(z) \frac{\partial w}{\partial z} \tag{1}$$

This is the Beltrami equation. If $\mu = 0$, it reduces to the Cauchy–Riemann equations ($\partial w / \partial \bar{z} = 0$, meaning the function is holomorphic). In general, however, $\mu \neq 0$.

Existence Proof (Grand Finale):

Deep theorems proved by heavy hitters on the analysis side of differential geometry, such as Morrey or Ahlfors–Bers, tell us that if $\mu$ behaves itself—as it does because our surface is regular—and $|\mu| < 1$, equation (1) locally has a homeomorphic solution $w$.

Writing this solution as $w = \xi + i \eta$, the pair $(\xi,\eta)$ gives the isothermal coordinates we seek, because in these coordinates the metric takes the form

$$ds^2 = \Lambda(\xi,\eta)(d\xi^2 + d\eta^2)$$

Thus $E = G$ and $F = 0$.

Conclusion:

Drawing an orthogonal, scaled grid on a surface turns out to amount to solving an equation in complex analysis. Geometry and Analysis... two halves of the same apple once again. I had to include rather more detail in this proof and briefly touch on things we are still far from seeing, but I hope everything is clear. I hope you remember the Cauchy–Riemann equations from complex analysis. If you are a physicist, there is nothing I can do; you chose this life.

We have gotten mathematics all over our hands and faces, but what has this theorem given us? If we write the First Fundamental Form as the matrix

$$I = \begin{bmatrix} E & F \\ F & G \end{bmatrix}$$

and, in accordance with the theorem above, choose a parametrization for which $E = G$ and $F = 0$, then $I$ can be represented as a diagonal matrix. This helps with calculations and also leads to something theoretically beautiful: local flatness. The coordinate system we call isothermal coordinates tells us that our surface locally resembles the same plane. Does that mean the flat-earthers are right in their own limited way? Let us not go that far.

Let us not forget this theorem; it will be useful when we define manifolds later. While we are on the subject, I also want to address harmonicity. We will need to define the Laplace–Beltrami operator for our surface. In its plainest form, the Laplace–Beltrami operator is the sum of the second derivatives taken on the surface. If we want to calculate the Laplacian (Beltrami, technically) of a function on our surface, we must take the sum of its second derivatives with respect to the surface parameters. You can see the elaborate detour I am taking to avoid saying “covariant derivative,” can you not—where is my special passport? Let us be a little formal and finally give the definitions.

Definition (Harmonic Function).

Consider a real-valued differentiable function $f : S \to \mathbb{R}$ defined on a surface $S$. If this function vanishes under the Laplace–Beltrami operator $(\Delta)$, which takes the geometry of the surface into account—that is, if

$$\Delta f = 0$$

then it is called a harmonic function.

What Does This Operator Look Like?

In our $E,F,G$ language, its explicit form is the following, where $W = \sqrt{EG - F^2}$ is the area element:

$$\Delta f = \frac{1}{W} \left[ \frac{\partial}{\partial u} \left( \frac{G f_u - F f_v}{W} \right) + \frac{\partial}{\partial v} \left( \frac{E f_v - F f_u}{W} \right) \right]$$

I would love to explain at length why these are the second derivatives involved, but we would wander too far afield. Let us approach the matter somewhat intuitively. In essence, we can say that the Laplace–Beltrami operator calculates averages of a function's values over the surface.

The Essence of the Matter (An Intuitive View):

My friend, for a function to be harmonic means that its value at a point equals the average of its values at neighboring points.

• Physically: The shape of a taut membrane or the temperature distribution on a plate in thermal equilibrium is harmonic.

• Geometrically: In isothermal coordinates $(E = G, F = 0)$, that complicated formula above reduces to

$$\Delta f = \frac{1}{E} (f_{uu} + f_{vv})$$

Thus “harmonicity” on the surface reduces to the familiar planar equation $(f_{xx} + f_{yy} = 0)$. This is why isothermal coordinates are magical: they make a twisty, curved surface feel like a flat sheet of paper on which we can do analysis.

Theorem

Isothermal coordinates $(u,v)$ are harmonic functions on the surface.

Proof

Let us settle this by an “old-school” calculation without going anywhere near Christoffel symbols, whatever those are. We have the general expression, in terms of $E,F,G$, for differential geometry's Swiss Army knife: the Laplace–Beltrami operator. For any function $\phi$, it is

$$\Delta \phi = \frac{1}{W} \left[ \frac{\partial}{\partial u} \left( \frac{G \phi_u - F \phi_v}{W} \right) + \frac{\partial}{\partial v} \left( \frac{E \phi_v - F \phi_u}{W} \right) \right]$$

Here, $W = \sqrt{EG - F^2}$ is our area element—that is, the determinant of the First Fundamental Form.

Now let us play our isothermal-coordinate trump card. What was our rule?

$$E = G = \lambda(u,v) \qquad \text{ve} \qquad F = 0$$

What does $W$ become in this case?

$$W = E$$

Now substitute these simplified values into that frightening formula above. It immediately becomes as docile as a lamb:

$$\Delta \phi = \frac{1}{E} \left[ \frac{\partial}{\partial u} \left( \frac{E\phi_u - 0}{E} \right) + \frac{\partial}{\partial v} \left( \frac{E\phi_v - 0}{E} \right) \right]$$

The factors of $E$ inside the brackets cancel. This is the magic of isothermal coordinates:

$$\Delta \phi = \frac{1}{E} \left[ \frac{\partial}{\partial u}(\phi_u) + \frac{\partial}{\partial v}(\phi_v) \right] = \frac{1}{E} (\phi_{uu}+\phi_{vv})$$

Now for the finishing move. We want to prove that the coordinates $u$ and $v$ are harmonic.

1. Check $u$ $(\phi = u)$:

$$u_u = 1, \qquad u_{uu} = 0$$ $$u_v = 0, \qquad u_{vv} = 0$$

Substitution gives

$$\Delta u = \frac{1}{E}(0+0)=0$$

2. Check $v$ $(\phi = v)$:

$$v_v = 1, \qquad v_{vv} = 0$$ $$v_u = 0, \qquad v_{uu} = 0$$

Substitution gives

$$\Delta v = \frac{1}{E}(0+0)=0$$ $$\square$$

We can in fact see from this that isothermal coordinates minimize the variations of a function on the surface. Yes, minimize. We can also see the following. Fix one parameter in isothermal coordinates and let the other vary. We are really drawing a level curve—physicists, lucky you, you understood a concept without my having to explain it. This curve gives the shortest path between two points on the surface; in other words, it is a geodesic. Let us explain this with a fine example: the Mercator projection.

Example 1.2.

Let the unit sphere $S^2$ be given. Using the standard geographic coordinates of latitude $\theta$ and longitude $\phi$, we parametrize it as

$$x(\theta,\phi) = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$$

Here, $0 < \theta < \pi$ and $0 < \phi < 2\pi$. In this form, the metric coefficients are $E = 1$ and $G = \sin^2\theta$. Thus $E \neq G$, so this is not an “isothermal,” angle-preserving map.

The Magic Touch:

Change the parameters as follows:

$$u = \ln \tan\left(\frac{\theta}{2}\right), \qquad v = \phi$$

Making this transformation and writing the parametrization $y(u,v)$—with a little trigonometric gymnastics—we get

$$x(u,v) = (\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u)$$

Now calculate the First Fundamental Form coefficients $(E,F,G)$ for this new parametrization $y$. Differentiating and taking inner products reveals the magnificent result

$$E = \langle x_u, x_u \rangle = \operatorname{sech}^2 u, \qquad F = 0, \qquad G = \langle x_v, x_v \rangle = \operatorname{sech}^2 u$$

Conclusion:

As you can see, $E = G = \lambda^2(u,v) = \operatorname{sech}^2 u$ and $F = 0$. Our friend Mercator made the sphere isothermal by stretching the latitude lines with a logarithmic function. That is why Greenland appears as large as Africa on Google Maps: preserving angles comes at the expense of area.

Example (Lagrange (1779): Every Surface of Revolution Shall One Day Be Isothermal!).

Only the sphere? No! Our friend Lagrange said, “Every surface created by rotating a function about an axis—that is, every surface of revolution—can be made isothermal.”

Parametrize a general surface of revolution by

$$x(\theta,\phi) = (f(\phi)\cos\theta, f(\phi)\sin\theta, g(\phi))$$

Here, the surface is built with the parameter $\phi$ along the profile curve and the parameter $\theta$ as the angle of rotation. Ordinarily, its metric has the rather messy coefficients

$$E = f(\phi)^2$$

and

$$G = (f')^2 + (g')^2$$

The Magic Touch (Lagrange's Method):

Change the parameters. Keep $u = \theta$, but define the following integral for the other parameter:

$$v = \int \frac{ \sqrt{(f'(\phi))^2 + (g'(\phi))^2} }{ f(\phi) } \, d\phi$$

Solving this integral and using the inverse function theorem to write a new parametrization $(y(u,v))$, the coefficients of the First Fundamental Form become

$$E = G = f^2(v) \qquad (\text{Burada } f \text{ aslında } f \circ h^{-1} \text{ dir}), \qquad F = 0$$

Conclusion:

Once again, we have achieved $E = G$ and $F = 0$. Thus even a vase, a funnel, or a bagel can be viewed as “graph paper in the plane” after a suitable integral transformation.

Of course, this was a specific example, and we have already proved something much better. Go play in the sand, Lagrange.

Harmonicity and Geodesics: Nature's Shortcuts

With such beautiful metrics satisfying $E = G$ at hand, let us address two important properties.

Why Are These Coordinates Harmonic?

As we proved in the previous section, in isothermal coordinates $(u,v)$ the Laplace–Beltrami operator gives

$$\Delta u = 0 \qquad \text{ve} \qquad \Delta v = 0$$

Consider the Mercator example $(E = G = \operatorname{sech}^2 u)$. The metric coefficients depend only on $u$. Our Laplace operator becomes

$$\Delta \psi = \frac{1}{E} (\psi_{uu} + \psi_{vv})$$

If we take $\psi = u$, then $u_{uu} = 0$ and $u_{vv} = 0$, so the result is immediately zero. Thus the latitude and longitude lines on the Mercator map are mathematically “harmonic” functions. They minimize tension on the surface.

Why Are Constant-Parameter Curves Geodesics?

Without getting too deep into the calculations, let me explain why constant-coordinate curves determine shortest paths here. Our question is: Are those straight vertical lines on the Mercator map—the meridians—really “shortest paths” on the sphere?

The answer is a resounding YES. But why?

The Power of Symmetry (The Mirror Test)

Think of surfaces of revolution: a sphere, cylinder, vase, or funnel. Each is formed by rotation about an axis. For our metric

$$ds^2 = \lambda(u)(du^2 + dv^2)$$

to depend only on $u$ (latitude) and not on $v$ (longitude/the angle of rotation) means that rotating the surface about itself does not change its geometry.

Now think of a meridian as a vertical line on the map. This line divides the surface exactly in half, like a slice through an orange.

• If you look to the right and left of this line, the geometry is perfectly symmetric.

• As an object travels along this line, it has no reason to go either right or left, because the two sides are mirror images of one another.

• Conclusion: The object continues straight ahead without deviating. To travel “without deviating” is precisely to follow a geodesic.

For this reason, all meridians on a surface of revolution—the ones running up and down—are automatically geodesics. No equation is needed; symmetry is enough!

Mercator, Airplanes, and Great Circles

Now picture the Mercator map, the one on which Greenland looks enormous.

• On the map: Meridians (longitudes) are straight, parallel lines running from top to bottom.

• On the sphere: These lines are arcs running from the North Pole to the South Pole. Complete those arcs, and they form enormous circles passing through the center of the Earth. We call them great circles.

This is very important in aviation: (Editor's note: now that he is a pilot, of course he was going to mention it.)

An airplane flying from Istanbul to New York does not follow a straight line on the map. Such a line is called a rhumb line, and it makes the route longer. The airplane curves northward. Why? Because the shortest path on a sphere is a great-circle arc.

HOWEVER! If you fly due south from Istanbul—for example, to South Africa—the situation changes. Your route lies along a meridian.

1. On the sphere, this is a great circle—the shortest path.

2. On the Mercator map, it is a perfectly straight vertical line.

Here lies the magic of Mercator's isothermal coordinates: although straight lines on a map do not normally show the shortest path, vertical lines (constant longitudes) are straight on the map and also give the shortest path in the real world.

In Summary:

On a rotationally symmetric surface such as a sphere, if your metric is independent of the angle ($G_v = 0$, which we will discuss), an ant walking along those vertical lines perpendicular to the axis of rotation has no reason to veer right or left. It walks straight ahead. The path it walks is the shortest distance between two points.

Summary

I had actually expected to introduce curvature in this chapter, but I realized that there were a great many things I needed to explain first. All these concepts can be explained far more easily using curvature and metrics, but since that creates something of a chicken-and-egg dilemma, I wanted to explain them using the intuitions and methods we have developed so far. In the next chapter, I hope to introduce curvature and begin to touch, somewhat indirectly, on manifolds. Until then, take good care of yourselves.