Grassmannians and conformal structure on absolutes

TL;DR

This paper explores the geometry of grassmannians linked to hermitian forms to elucidate the relationship between projective geometries and conformal structures on their boundaries, unifying several classical geometries.

Contribution

It introduces a unified geometric framework connecting grassmannians with conformal structures on various space boundaries, extending classical models.

Findings

Unified description of conformal structures on space boundaries

Relation between grassmannians and boundary geometries

Application to hyperbolic, de Sitter, and anti-de Sitter spaces

Abstract

We study grassmannians associated with a linear space with a nondegenerate hermitian form. The geometry of these grassmannians allows us to explain the relation between a (pseudo-)riemannian projective geometry and the conformal structure on its ideal boundary (absolute). Such relation encompasses, for instance, the usual conformal structure on the absolute of real hyperbolic space, the usual conformal structure on the absolute of de Sitter space, the conformal contact structure on the absolute of complex hyperbolic space, and the causal structure on the absolute of anti-de Sitter space.