Regulous vector bundles

TL;DR

This paper introduces and studies regulous vector bundles in real algebraic geometry, establishing their properties, characterizations, and relationships with topological bundles, thereby advancing the understanding of regulous geometry.

Contribution

It defines regulous vector bundles, explores their algebraic and geometric properties, and characterizes them using curve-rational functions and stratified-algebraic structures.

Findings

Regulous vector bundles are characterized among families of vector spaces over affine regulous varieties.

They are identified with stratified-algebraic vector bundles.

Relationships between regulous and topological vector bundles are established.

Abstract

Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified-algebraic vector bundles. Furthermore, using new results on curve-rational functions, we characterize regulous vector bundles among families of vector spaces parametrized by an affine regulous variety. We also study relationships between regulous and topological vector bundles.