Rational real algebraic models of topological surfaces

TL;DR

This paper proves that each topological surface (nonorientable surface, sphere, or torus) has a unique rational real algebraic model, extending known results to all such surfaces.

Contribution

It establishes the uniqueness of rational real algebraic models for all topological surfaces previously known only for specific cases.

Findings

Each topological surface has exactly one rational real algebraic model.

The result extends earlier known cases to all surfaces of this type.

The proof relies on properties of rational real algebraic surfaces.

Abstract

Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the two-sphere, torus, projective plane and the Klein bottle.