## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 71, Issue 4 (2006), 1081-1432

### Computability results used in differential geometry

Barbara F. Csima and Robert I. Soare

#### 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.

#### Article information

**Source**

J. Symbolic Logic Volume 71, Issue 4 (2006), 1394-1410.

**Dates**

First available: 20 November 2006

**Permanent link to this document**

http://projecteuclid.org/euclid.jsl/1164060462

**Digital Object Identifier**

doi:10.2178/jsl/1164060462

**Mathematical Reviews number (MathSciNet)**

MR2275866

**Zentralblatt MATH identifier**

1109.03035

#### Citation

Csima, Barbara F.; Soare, Robert I. Computability results used in differential geometry. Journal of Symbolic Logic 71 (2006), no. 4, 1394--1410. doi:10.2178/jsl/1164060462. http://projecteuclid.org/euclid.jsl/1164060462.