Fried's theorem for boundary geometries of rank one symmetric spaces

TL;DR

This paper generalizes Fried's theorem to all boundary geometries of rank one symmetric spaces, showing that closed manifolds with similarity structures are either complete or have a developing map covering a punctured Heisenberg-type space.

Contribution

It extends Fried's theorem to a broader class of boundary geometries of rank one symmetric spaces, unifying the theory across different geometrical structures.

Findings

Closed manifolds with similarity structures are either complete or develop onto a punctured Heisenberg space.

The theorem applies to all boundary geometries of rank one symmetric spaces.

Provides a unified proof for various boundary geometries.

Abstract

After introducing the different boundary geometries of rank one symmetric spaces, we state and prove Fried's theorem in the general setting of all those geometries: a closed manifold with a similarity structure is either complete or the developing map is a covering onto the Heisenberg-type space deprived of a point.