Algebraic Smooth Structures 1

TL;DR

This paper introduces and studies algebraic smooth structures on semi-integral domains, establishing their uniqueness and constructing non-polynomial smooth functions without topology, with applications to smooth manifold mappings.

Contribution

It defines C-structures and smooth structures algebraically, proves uniqueness on semi-integral domains, and constructs novel smooth functions, extending the theory without topological methods.

Findings

Semi-integral domains not being fields admit unique smooth structures.

Constructed non-polynomial smooth functions on semi-integral domains.

Demonstrated a correspondence between algebra homomorphisms and smooth maps between manifolds.

Abstract

In this paper which is the first of a series of papers on smooth structures, the concepts of C-structures and smooth structures are introduced and studied. The notion of smooth structure on semi-integral domains is given. It is shown that each semi-integral domain which is not a field, admits a unique smooth structure and a large class of non-polynomial smooth functions on some semi-integral domains is constructed. A smooth function from Z-{0} into Z is given which does not extend to a smooth function on Z. No concept from topology is used. As an application, it is shown that: Theorem - Let M and N be finite dimensional smooth manifolds. The algebra of real smooth functions on M (resp. N) will be denoted by A (resp. B). Assume that T is a homomorphism from B into A. Then, there exists exactly one smooth mapping f from M into N such that T=f*.