Geometry & Topology

A geometric transition from hyperbolic to anti-de Sitter geometry

Jeffrey Danciger

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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 m5.

Article information

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

Received: 25 February 2013
Accepted: 26 June 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53B30: Lorentz metrics, indefinite metrics 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

geometric transition hyperbolic AdS cone manifold tachyon projective structure transitional geometry half-pipe geometry


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.

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