Computability Theory and Differential Geometry

LetMbe a smooth, compact manifold of dimensionn≥ 5 and sectional curvature ∣K∣ ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics onMmodulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machineTe,eϵ ω, there is a sequence (uniformly effective ine) of homologyn-sphereswhich are also hypersurfaces, such thatis diffeomorphic to the standardn-sphereSn(denoted)iffTehalts on inputk, and in this case the connected sum, so, andis associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time ae σe(k)ofTeon inputsy<k.At their request Soare constructed a particular infinite sequence {Ai}ϵωof c.e. sets so that for allithe settling time of the associated Turing machine forAidominates that forAi+1, even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a “fractal” like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as “the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization.” From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structuralcomplexityof the geometry and topology, not merelyundecidabilityresults as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they usenontrivialinformation about c.e. sets, the Soare sequence {Ai}iϵωabove, not merely Gödel's c.e. noncomputable set K of the 1930's; and (3)withoutusing computability theory there is no known proof that local minima exist even for simple manifolds like the torusT5(see §9.5).