Book on manifolds from a sheaf-theoretic/locally ringed space PoV 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.
 A: I'm not very familiar with this book (in particular, I don't know how introductory or not it is), but I think

Torsten Wedhorn, Manifolds, Sheaves, and Cohomology. Springer Studium Mathematik—Master. Springer Spektrum, Wiesbaden, 2016. xvi+354 pp. ISBN: 978-3-658-10632-4; 978-3-658-10633-1

would fit your description.
A: Ramanan's "Global Calculus" is a very nice introductory text which defines manifolds by their sheaves of differentiable functions. I don't know if this is what you're looking for: I don't see it as very algebraic, and tools from analysis have an important role.
A: Jet Nestruev (a collective author, I think) does this in "Smooth Manifolds and Observables".
A: 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.
