Smooth and F--smooth systems

TL;DR

This paper reviews the geometric theory of smooth and F-smooth systems of maps, sections, and connections on fibred manifolds, highlighting their structures and interrelations.

Contribution

It provides a comprehensive review of smooth and F-smooth systems of maps, sections, and connections within the geometric framework of fibred manifolds.

Findings

Unified geometric framework for smooth systems

Extension to F-smooth systems based on Fr"olicher's concept

Clarification of relationships between different smooth systems

Abstract

We review the geometric theory of \emp{smooth systems of smooth maps}, of \emp{smooth systems of smooth sections} of a smooth double fibred manifold and of \emp{smooth systems of smooth connections} of a smooth fibred manifold. Moreover, after reviewing the concept of \emp{F-smooth space} due to A. Fr\"olicher, we discuss the \emp{F-smooth systems of smooth maps}, the \emp{F-smooth systems of fibrewisely smooth sections} of a smooth double fibred manifold, the \emp{F-smooth systems of fibrewisely smooth connections} of a smooth fibred manifold and of \emp{F-smooth connections} of an F-smooth system of fibrewisely smooth sections.