Projection theorems in hyperbolic space

TL;DR

This paper extends projection theorems to hyperbolic space, providing geometric insights into the behavior of projections and characterizations of unrectifiable sets in this non-Euclidean setting.

Contribution

It introduces Marstrand-type projection theorems and a Besicovitch-Federer characterization for hyperbolic space, advancing geometric measure theory in non-Euclidean geometry.

Findings

Established projection theorems for hyperbolic space

Characterized unrectifiable sets via hyperbolic projections

Provided geometric proofs for these theorems

Abstract

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of hyperbolic $n$-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.