A primer on the (2 + 1) Einstein universe

Abstract

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.

On the restricted conformal group of the (1+n)-Einstein static universe

“…( Here ξ * stands for the fundamental vector field on Q(M) generated by ξ. 6 A form is semibasic of it annihilates the vertical vector fields. Theorem 2 (S. Kobayashi).…”

“…The (1 + n)-Einstein static universe is the product space E 1,n ∼ = R× S n with the Lorentz product metric −dt 2 + g S n , where g S n denotes the standard metric of S n (cf. [6,15,16]). The space E 1,n is the universal covering of both E 1,n I and E 1,n II .…”

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

“…We first exhibit a natural parametrization based on which we can define a symplectic arc length. We briefly discuss the standard conformal structure of the Grassmannian of the oriented Lagrangian planes of R 4 (we refer to [2], [12] for more details). Subsequently we define the osculating curve and the phase portraits of a Lagrangian curve.…”

“…Since the notion of null curve is invariant by conformal transformations, the natural environment is the conformal compactification of the Minkowski 3-space, which can be thought of as the manifold of all oriented Lagrangian vector subspaces of R 4 . Such a link between symplectic and conformal geometry is specific to the four-dimensional case.The reason lies in the fact that Sp(4, R) is a covering group of the connected component of the identity of O (3,2). We begin with a brief description of the conformal structure of the Grassmannian of oriented Lagrangian planes in R 4 .…”

Critical Robertson–Walker universes

“…Despite their old history [16,24,38,43], RW cosmological models still provide a valuable testing ground for new ideas and theories. These include gravitational thermodynamics [9,40,46], relativistic diffusion [2], cyclic and conformal cyclic cosmology [3,36,37,41,42], Lorentz conformal geometry [1,4,15], super-symmetry and string theory [7,39].…”