## Geometry & Topology

### A geometric transition from hyperbolic to anti-de Sitter geometry

Jeffrey Danciger

#### Abstract

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti-de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the “other side” of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these methods in the case when the manifold is the unit tangent bundle of the $(2,m,m)$ triangle orbifold for $m≥5$.

#### Article information

Source
Geom. Topol., Volume 17, Number 5 (2013), 3077-3134.

Dates
Accepted: 26 June 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732695

Digital Object Identifier
doi:10.2140/gt.2013.17.3077

Mathematical Reviews number (MathSciNet)
MR3190306

Zentralblatt MATH identifier
1287.57020

#### Citation

Danciger, Jeffrey. A geometric transition from hyperbolic to anti-de Sitter geometry. Geom. Topol. 17 (2013), no. 5, 3077--3134. doi:10.2140/gt.2013.17.3077. https://projecteuclid.org/euclid.gt/1513732695

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