The next example comes from more classical differential geometry, namely the study of minimal surfaces. It is well known that the shortest path between two points on the plane is the straight line which joins them. Substituting the plane for a curved surface, the shortest path is now along a geodesic. Geodesics are curves which satisfy a second-order ordinary differential equation. Indeed, they correspond to Newton's equation for a particle moving freely on the surface, subject to no other force than the one keeping it to the surface.
There is a natural higher-dimensional analogue to a geodesic; that of a minimal surface or, more generally, minimal submanifold. We can understand the surface or the submanifold as that traced by a string or a higher-dimensional extended object (branes, in a string theory context) which is moving in a space. Minimal submanifolds are those for which the volume does not change under infinitesimal (normal) deformations. In other words, the volume is extremal, but not necessarily minimal as their (historical) name would suggest. A typical example are soap bubbles, whose tension is proportional to the area, and hence try to configure themselves in a way which minimises area. Again, the condition of being a minimal surface translates into a second-order differential condition on the embedding.
There is often a first-order condition which implies this second-order condition. Let's go back to the straight line minimising distance in the plane. In order to follow a straight line, it is not necessary to solve a second-order equation: it is enough to ensure that the direction does not change. This is a zeroth order condition on the direction, which is the derivative along the path, whence it is a first-order condition. To some extent all we have done is integrate the second-order equation ẍ = 0 to the first-order equation ẋ = constant, which — I will admit — seems a little trivial.
This idea has a nontrivial generalisation, however, in the form of a calibration, a notion introduced by Harvey and Lawson.
Recall that a p-form is a smooth assignment of a number to every tangent p-plane. A calibration is a closed p-form ω on a manifold such that
for every tangent p-plane Π, where dvol is the volume form constructed from the metric. A plane is said to be calibrated (by ω) if the above bound is saturated. A p-dimensional submanifold is said to be calibrated if all its tangent spaces are calibrated. Clearly this is a first-order condition, as it is linear on the tangent plane.
A calibration ω defines certain "faces" on the grassmannian of p-planes consisting of those planes which it calibrates. The condition of a submanifold being calibrated is thus equivalent to saying that its tangent planes are restricted to one such face of the grassmannian. In the case of curves, the faces are points, hence the fact that geodesics have "constant" direction.
The fundamental lemma in the theory of calibrations is that a calibrated submanifold is volume-minimising in its homology class. My friend Michael Singer can't help bemoaning the fact that no mention of calibrations, however cursory, ever omits the one-line proof of this fact. Well, we couldn't disappoint him, could we? So let N be a calibrated submanifold and N' another submanifold in its homology class. Then
where the first equality is a consequence of N being calibrated, the second is Stokes', and the last equality holds because ω is a calibration.
Where is the supersymmetry?
Nunca perseguí la gloria,
ni dejar en la memoria
de los hombres mi canción;
yo amo los mundos sutiles,
ingrávidos y gentiles,
como pompas de jabón.
— Antonio Machado
Proverbios y Cantares I