We develop the Lorentzian geometry of a crooked halfspace in 2 + 1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parallelism classes of timelike lines, or particles, in a crooked halfspace is a geodesic halfplane in the hyperbolic plane. Every point in an open crooked halfspace lies on a particle. The correspondence between crooked halfspaces and halfplanes in hyperbolic 2-space preserves the partial order defined by inclusion, and the involution defined by complementarity. We find conditions for when a particle lies completely in a crooked half space. We revisit the disjointness criterion for crooked planes developed by Drumm and Goldman in terms of the semigroup of translations preserving a crooked halfspace. These ideas are then applied to describe foliations of Minkowski space by crooked planes.Date: November 7, 2018. 2000 Mathematics Subject Classification. 53B30 (Lorentz metrics, indefinite metrics), 53C50 (Lorentz manifolds, manifolds with indefinite metrics).
Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface Σ. We classify such complete affine structures when Σ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Σ, the deformation space identifies with two opposite octants in R 3 . Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4, Z).
The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension 2 + 1, in which there is a rich interplay with symplectic geometry.
A Margulis spacetime is a complete affine 3‐manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic to a two‐holed cross‐surface Σ, that is, the complement of two disjoint disks in ℝP2. We show that every such manifold is homeomorphic to a solid handlebody of genus 2, and admits a fundamental polyhedron bounded by crooked planes. Furthermore, the deformation space is a bundle of convex four‐sided cones over the space of marked hyperbolic structures. The sides of each cone are defined by invariants of the two components of ∂ Σ and the two orientation‐reversing simple curves. The two‐holed cross‐surface, together with the three‐holed sphere, are the only topologies Σ for which the deformation space of complete affine structures is finite‐sided.
A Margulis spacetime is a complete flat Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface Σ homotopy-equivalent to M . The purpose of this paper is to classify Margulis spacetimes when Σ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of Σ. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces. This extends our previous work on affine deformations of three-holed sphere and two-holed cross surfaces. arXiv:1501.04535v1 [math.DG] 19 Jan 2015 45 10.2. Geometric tameness 45 References 47 2. Notation and terminology 2.1. Elementary conventions. By a triple (respectively pair ) we shall always mean an ordered triple or pair respectively, unless otherwise stated.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.