Smooth Manifolds (Algebraic Definition)

Abstract

Using the definitions of the notion of point and those of the commutative algebras under consideration, the formal algebraic definition of a smooth manifold is given in this chapter. It is defined as the dual space $|\mathcal F|$ of any complete geometric algebra $\mathcal F$, supplied with an open covering $\{U_k\}$ in the Zariski topology such that each algebras $\mathcal F|_{U_k}$ is isomorphic to $C^{\infty }(U_k)$. Numerous concrete examples of how this works are presented.