What is curvature? (25.04.2020)

Parvaneh Joharinad and Jürgen Jost

Traditionally, curvature is ascribed to smooth curves or surfaces that are not straight or flat, and smooth and curved shapes can appear very beautiful. While aesthetically pleasing, such a concept of curvature is not fully satisfactory from a mathematical perspective. First of all, it conflates intrinsic and extrinsic aspects. If you roll a sheet of paper into a cylinder, it becomes curved in that sense although its intrinsic geometry, i.e., the lengths of curves drawn on it and the angles between such curves do not change. On the other hand, a surface like a sphere is intrinsically different from a plane, because it cannot be mapped onto the plane without distorting lengths or angles. Thus, such a concept of curvature does not distinguish between the cylinder that has the same intrinsic geometry as the plane, and the sphere which is intrinsically different. Secondly, computing the traditional curvature requires taking second derivatives, but not all geometric objects are smooth.

The first problem was overcome by Gauss and Riemann who developed a purely intrinsic concept of curvature. Riemann's work, in particular, made it possible to conceive of curved space-times in Einstein's general theory of relativity. In fact, a fundamental insight of Riemann was that the geometry of a smooth manifold with a metric structure can be completely characterized by its curvature tensor, and also by the sectional curvatures derived from that tensor. Thus, concerning the second problem, one should look for characteristic relations for the geometry of general metric spaces. In a metric space \((X,d)\), the distances \(d(x_i,x_j)\) between three points \(x_1,x_2,x_3\) need to satisfy the triangle inequality \(d(x_i,x_k)\le d(x_i,x_j)+d(x_j,x_k)\) for \(i,j,k \in \{1,2,3\}\). But the geometry gets constrained when we look at the distances from these three points to other points in \(X\). We now have probably achieved the most general form by determining curvature-like quantities in general metric spaces by such relations. We determine radii \(r_i\)with \(r_i+r_j=d(x_i,x_j)\), \(1\leq i<j\leq3\) and put

\begin{equation*} \rho(x_1,x_2,x_3):=\inf_{x\in X}\max_{i=1,2,3}\frac{d(x_i,x)}{r_i}. \end{equation*}

The figure shows the situation in the Euclidean plane. The three closed disks on the left with the radii \(r_i\) intersect in pairs, but don't have a triple intersection. To get a triple intersection, they must be enlarged by the factor \(\rho\), as shown by the red circles on the right. In any metric space, this \(\rho\) lies between \(1\) and \(2\). In Riemannian manifolds, the larger \(\rho\), the larger the curvature. \(\rho=1\), for example, is realized on trees, where trees are mathematically defined as graphs that keep branching out, or more generally in so-called hyper-convex spaces, in which any collection of pairwise intersecting balls also has a common intersection. In the Riemannian context, this corresponds to the borderline case where the curvature becomes \(-\infty\). The case \(\rho=2\), on the other hand, is realized, for example, if we consider three equidistant points on a circular line as metric space. Such a \(\rho\) thus describes the qualitative geometrical properties of metric spaces, and it can also be evaluated well on empirical data, where sets of point with distances are given. From this perspective, curves are actually not curved at all, because they can be straightened without changing their length. The essence of curvature is rather the extent to which the size of balls intersecting in pairs in a metric space has to be increased to get a common intersection.

References

[1]   Joharinad, P. Jost, J. (2019), Topology and curvature of metric spaces, Adv.Math. 356, article 106813.
[2]   Joharinad, P. Jost, J. (2020), Topological representation of the geometry of metric spaces, arXiv:2001.10262

04.05.2020, 16:09