Computability results used in differential geometry

Abstract

Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating properties. Although these computability results had been announced earlier, their proofs have been deferred until this paper.

Computably enumerable sets have long been used to prove undecidability of mathematical problems such as the word problem for groups and Hilbert’s Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima.