Let M be a smooth. compact manifold of dimension n greater than or equal to 5 and sectional curvature K less than or equal to 1. Let Met(M) = Riem(M)/ Diff(M) be the space of Riemannian metrics on M modulo isometrics. 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 machine T-e e epsilon omega. there is a sequence (uniformly effective in e) of homology n-spheres {P-k(e)}(kepsilonomega) which are also hypersurfaces. such that P-k(e) is diffeomorphic to the standard n-sphere S-n (denoted P-k(e) approximate to(diff) S-n) iff T-e halts on input k. and in this case the connected sum N-k(e) = M#P-k(e) approximate to(diff) M . so N-k(e) epsilon Met(M). and Nek is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time sigma(e)(k) of T-e on inputs y < k.At their request Soare constructed a particular infinite sequence {A(i)}(epsilonomega) of c.e. sets so that for all i the settling time of the associated Turing machine for A(i) dominates that for A(i+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 structural complexity of the geometry and topology. not merely undecidability results as in the word problem for groups. Hilbert's Tenth Problem. or most other applications: (2) they use nontrivial information about c.e. sets. the Scare sequence {A(i)}(iepsilonomega) above. not merely Godel's c.e. noncomputable set K of the 1930's: and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T-5 (see 9.5).
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机译：令M平滑。尺寸n大于或等于5且截面曲率 K 小于或等于1的紧凑流形。令Met（M）= Riem（M）/ Diff（M）为M模等轴测图上黎曼度量的空间。 Nabutovsky和Weinberger研究了Met（M）上某些功能（例如直径）的子级集（和局部极小值）的连通分量。他们表明，对于每台图灵机，T-e e epsilonΩ。有一个序列（在e中均有效）同源n球{P-k（e）}（kepsilonomega），它们也是超曲面。这样，当输入k停止时，P-k（e）与标准n球S-n（表示为P-k（e）近似于（diff）S-n）微分。在这种情况下，连接和N-k（e）= M＃P-k（e）近似于（diff）M。所以N-k（e）epsilon Met（M）。 Nek与Met（M）上的直径函数的局部最小值相关联，该函数的深度大致等于Te在输入y 展开▼
