Projection Theorems and Isometries of Hyperbolic Spaces

TL;DR

This paper establishes a restricted projection theorem for a specific family of projections related to hyperbolic space isometries, connecting geometric analysis with Lie group representations.

Contribution

It introduces a new projection theorem for a family of projections arising from the adjoint representation of a unipotent subgroup in hyperbolic space.

Findings

Proves a restricted projection theorem for an (n-2)-dimensional family of projections.

Connects geometric projection results with Lie algebra representations of hyperbolic isometries.

Provides tools potentially useful for geometric measure theory in hyperbolic spaces.

Abstract

We prove a restricted projection theorem for an n-2 dimensional family of projections from $\mathbb R^n$ to $\mathbb R$. The family we consider arises naturally in the context of the adjoint representation of the maximal unipotent subgroup of ${\rm SO}(n-1,1)$ on the Lie algebra of ${\rm SO}(n,1)$.