About Curvature

What is curvature? We have an intuitive notion that some curves are curvier than others, so how have mathematicians sorted this out?

I have been reading The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions by Shing-Tung Yau and Steve Nadis. It turns out that the notion of curvature, and in particular, Ricci curvature, is fundamental to all the considerations therein. From my deeply shallow and mostly forgotten studies of general relativity I recalled that the Ricci tensor was (a) an index contracted Riemann curvature tensor and (b) an essential component of the Einstein tensor. Neither bit of intellectual flotsam gave me any significant insight into what Ricci curvature really was.

For me to understand something, I need to have a mental picture that can be expressed in familiar notions. The simplest notion of curvature is that we associate with a circle. We have an intuitive notion that a smaller circle is “more curved” than a larger one. We can make this notion precise in terms of the reciprocal of the radius of the circle – if the circle has radius R, its curvature is 1/R.

So what about ellipses, hyperbolas, and random curvy lines, still operating in the Euclidean plane? Again, we can see that this kind of curve, unlike a circle, seems to be more curved in some places than others. Can we fit a circle inside such a curve? It turns out that there are infinitely many curves tangent to a given plane curve at a point, but we can find a “best” one – the so called Osculating (or kissing) circle – the circle that stays closest to the curve near the point in question. I should mention that we are sticking to sufficiently smooth curves here, that is, curves without any corners or gaps – (twice continuously differentiable, to be technical). The curvature of that plane curve is then defined to be the reciprocal of the radius of that kissing circle.

Things get a bit more complicated if we go to three (or more) dimensions. Suppose we have a curve embedded in a two dimensional surface – one like the center of a saddle, for example. Imagine yourself with an ant’s eye view from the center of the saddle (or a person’s eye view from the top of a mountain pass). If you look one direction, the world curves up, but in another, it curves down. It’s obvious that lots of circles could be fitted – and a sphere that fit well in one direction, wouldn’t fit in the other.

So does such a surface need infinitely many “curvatures” to describe it, one for each direction? Fortunately not, if the surface is sufficiently smooth. If so, two principal curvatures suffice. At each point of the surface, one can draw a normal line (or perpendicular) to the surface. If we now imagine cutting the surface with planes containing the normal in every possible way, each plane will slice the surface into a unique plane curve which will have an osculating circle with a center on the normal line. Count curvatures as positive if they lie on one side of the surface (the inside for closed surfaces like spheres) and negative if on the other. The maximum such curvature is one principal curvature and the minimum, the other. Their product is the famous Gaussian curvature. The most remarkable thing about the Gaussian curvature (see link) is the fact that it turns out to depend only on the way distances are measured on the surface, not upon the way the surface is embedded in higher dimensional space. This result is one of the foundational notions of differential geometry.

Are we there (to Ricci curvature) yet? Not quite, but another gas station or two ought to do it.

As we go to still higher dimensions, more ways to curve become possible. Recall that for the two–dimensional surface we took a bunch of planar slices through the surface to get the two principal curvatures and the Gaussian curvature. We can do something analogous in higher dimensions, though for technical reasons we need to operate in the tangent space and its exponential map – concepts which I won’t try to explain – but try to think of the operation as a kind of best approximation to slicing up the space itself. Each slice is a bit of 2-D (2–dimensional) surface, and the sectional curvature is the Gaussian curvature of that 2-D surface. If we know all the sectional curvatures at a point, we know all about the curvature at that point.

Consider the unit vectors tangent to the manifold at a point (unit vectors in the tangent space). For each such unit vector in n dimensional space, there is an n-2 dimensional family of planes containing that unit vector. If we compute the average sectional curvature for all the planes containing the unit vector A, we get Ricci(A,A), the Ricci curvature in direction A. Since the Ricci tensor Ricci(A,B) is bilinear and symmetric, if we know Ricci(A,A) for all A, we can compute Ricci(A,B) for all A and B.

Needless to say, I have left out a lot. I haven't made clear the distinction between intrinsic and extrinsic curvature, and I haven't mentioned that you really need the full Riemann tensor in more than three dimensions. I'm counting on the mathematically literate among my readers to catch the more egregious errors.