Dimensions of projections of sets on Riemannian surfaces of constant curvature

TL;DR

This paper extends classical projection theorems like Marstrand's to Riemannian surfaces of constant curvature, providing dimension distortion estimates for orthogonal projections.

Contribution

It adapts and applies the theory of Peres and Schlag to derive projection theorems on curved surfaces, a novel extension beyond Euclidean spaces.

Findings

Derived Hausdorff dimension distortion estimates for projections on curved surfaces.

Established versions of Marstrand, Kaufman, and Falconer theorems in Riemannian geometry.

Provided tools for analyzing fractal sets on curved manifolds.

Abstract

We apply the theory of Peres and Schlag to obtain estimates for generic Hausdorff dimension distortion under orthogonal projections on simply connected two dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand theorem, Kaufman's theorem and Falconer's theorem in the above geometrical settings.