Many years ago I spent several months obsessed with a marvellous theorem of Lohkamp, without any new theorems of my own resulting. I have learnt both mathematical and moral lessons from this, which I explore in this posting (and possible sequels).
Theorem (Lohkamp): Let be a Riemannian manifold whose dimension is at least three. Then there is a Riemannian metric on , which can be taken to be arbitrarily close to in the metric, so that has negative Ricci curvature.
Three questions come to mind:
- Why does this need dimension at least three?
- The corresponding result for positive Ricci curvature is false. Why the asymmetry?
- For what other curvature conditions can we prove such a result?
The third question amounts to asking what curvature properties are natural properties of the underlying metric space, or the underlying metric measure space. Both positive and negative sectional curvatures, more generally lower and upper bounds on sectional curvatures, can be expressed in terms of comparison triangles. This makes them metric properties. More recently it has been shown that lower bounds on Ricci curvature can be expressed in terms of so called optimal transport on metric measure spaces.
On the other hand, Lohkamp’s theorem shows that negative Ricci curvature is not an interesting natural property of metric measure spaces. More precisely, the set of Riemannian metrics with negative Ricci curvature is dense in the space of all Riemannian metrics – in particular there are no topological restrictions. In general, we would like to know which properties are dense. In particular there are no topological restrictions.
One can, to some extent, answer the first two questions relatively easily, but the answers are illuminating. I return to these in sequels. For now, I will just end with a moral from my previous failure.
Moral: We cannot avoid hard analysis. My previous attempts all amounted to attempting to reduce the problem to understanding the dominant term. However one (presumably) cannot avoid the stubborn persistence of a pair of comparable terms of opposite signs, one only slightly larger than the other. In this case, and many others that also are hard to intuitively understand, qualitative properties are determined by quantitative behaviour of the underlying system.