\(C^\infty\)-differentiable spaces
type: Item
instance of: scholarly article
title: \(C^\infty\)-differentiable spaces
zbMATH Open document ID: 1039.58001
author: Juan B. Sancho de Salas
author: Juan A. Navarro González
published in: Lecture Notes in Mathematics
publication date: 2003-12-03T00:00:00Z
review text: Differential geometry is traditionally regarded as the study of smooth manifolds, but sometimes this framework is too restrictive since it does not admit certain basic geometric constructions. The study of differentiable spaces allows including these constructions. For example, elementary surfaces of classical geometry such as a quadratic cone or a doubly counted plane are not smooth manifolds, but they have a natural differentiable structure defined by means of the consideration of an appropriate algebra of differentiable functions. In this monograph, the authors develop the theory of differentiable spaces. For the sheaf \(C^\infty(\mathbb R^ n)\) of differentiable functions of class \(C^\infty\) and some closed ideal \(I\) with respect to the Fréchet topology of uniform convergence on compact sets of functions, the quotient \(C^\infty(\mathbb R^ n)/I\) can be regarded as an algebra of differentiable functions on the closed subset \((I)_ 0=\{x;\;f(x)=0,\;\forall f\in I\}\) of \(\mathbb R^ n\) and is called a differentiable algebra. If \(I\) is the ideal of all \(C^ \infty\)-functions vanishing on a given closed set \(X\subset \mathbb R^ n\), then the quotient algebra \(C^\infty(\mathbb R^ n)/I\) is identified with a ring of real valued functions on \(X\) and is called a reduced differentiable algebra. Each differential algebra \(A\) can be replaced by a ringed space \(\text{Spec}_ r(A)\), a topological space with a sheaf of rings, called the real spectrum of \(A\). A ringed space is said to be affine differentiable space if it is isomorphic to the real spectrum of some differentiable algebra. A ringed space in which each point has an open neighborhood that is an affine differentiable space is called a differentiable space. The manuscript comprises eleven chapters and two appendices. Chapter 1 presents the elementary theory of smooth manifolds in the spirit of differentiable spaces. In chapter 2, the authors study differential algebras to introduce the definition an…
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 58A40
Mathematics Subject Classification ID: 13J99
zbMATH DE Number: 2012370
zbMATH Keywords: differentiable algebras
zbMATH Keywords: differentiable space
zbMATH Keywords: ringed space
zbMATH Keywords: convex module
zbMATH Keywords: fibred product
zbMATH Keywords: finite morphism
zbMATH Keywords: sheaf
zbMATH Keywords: \(r\)-jet
reviewed by: Andrew Bucki
MaRDI profile type: Publication
full work available at URL: b13465
OpenAlex ID: W4242340237
DOI: 10.1007/B13465
description: scientific article; zbMATH DE number 2012370