## Bulletin of Symbolic Logic

- Bull. Symbolic Logic
- Volume 10, Issue 4 (2004), 457-486.

### Computability theory and differential geometry

#### Abstract

Let $M$ be a smooth, compact manifold of dimension $n\geq 5$ and sectional curvature $ |K| \leq 1$. Let Met($M$) = Riem($M$)/Diff($M$) be the space of Riemannian metrics on $M$ modulo 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 machine $T_e, e \in \omega$, there is a sequence (uniformly effective in $e$) of homology n-spheres {$P_k^e$}$_ k\in\omega$ which are also hypersurfaces, such that $P_k^e$ is diffeomorphic to the standard $n$-sphere $S^n$(denoted $P_k^e \approx_{\rm{diff}} S^n$ iff $T_e$ halts on input k, and in this case the connected sum $N_k^e = M \# P_k^e \approx_{\rm{diff}}M$, so $N_k^e \in $Met($M$), and $N_k^e$ 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 { $A_i$ }$_{\in
\omega}$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 Soare sequence {$A_i$ }$_{\in
\omega}$above, not merely Gödel'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).

#### Article information

**Source**

Bull. Symbolic Logic, Volume 10, Issue 4 (2004), 457-486.

**Dates**

First available in Project Euclid: 3 December 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.bsl/1102083758

**Digital Object Identifier**

doi:10.2178/bsl/1102083758

**Mathematical Reviews number (MathSciNet)**

MR2136634

**Zentralblatt MATH identifier**

1085.03033

#### Citation

Soare, Robert I. Computability theory and differential geometry. Bull. Symbolic Logic 10 (2004), no. 4, 457--486. doi:10.2178/bsl/1102083758. https://projecteuclid.org/euclid.bsl/1102083758