TITLE: Book on manifolds from a sheaf-theoretic/locally ringed space PoV
QUESTION [11 upvotes]: I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds from the differentiable viewpoint; I wonder if a text introducing differential manifolds from an algebraic viewpoint exists.

REPLY [4 votes]: In Introduction to differential geometry (see the review) by R.Sikorski the author introduces the concept of (what is now called) Sikorski space. Sikorski spaces are "affine, reduced differential spaces" and hence they can be approached algebraically by looking at their coordinate rings. Differentiable manifolds are important examples Sikorski spaces. Unfortunately the book was not translated to english. Luckily there are some publications (in english and perhaps in french) by Sikorski in which he explains this very natural concept. One of them is Differential modules.    
A book $C^{\infty}$ Differentiable Spaces by Navarro González and Sancho de Salas develop theory of differentiable spaces by first constructing real spectra for smooth algebras and then glue them in order to obtain general spaces. This is analogical to the development of algebraic geometry (scheme theory) by Grothendieck and his school. The book might be a bit advanced.