linguistic:paraphrasing, linguistic:voice_change
Sheaves of local rings

A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space. This approach is strongly influenced by the theory of schemes in algebraic geometry, but uses local rings of the germs of differentiable functions. It is especially popular in the context of complex manifolds.

We begin by describing the basic structure sheaf on R. If U is an open set in R, let

consist of all real-valued k -times continuously differentiable functions on U. As U varies, this determines a sheaf of rings on R. The stalk O for p ∈ R consists of germs of functions near p, and is an algebra over R. In particular, this is a local ring whose unique maximal ideal consists of those functions that vanish at p. The pair (R, O) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings.

A differentiable manifold (of class C) consists of a pair (M, O) where M is a second countable Hausdorff space, and O is a sheaf of local R -algebras defined on M, such that the locally ringed space (M, O) is locally isomorphic to (R, O). In this way, differentiable manifolds can be thought of as schemes modeled on R. This means that for each point p ∈ M, there is a neighborhood U of p, and a pair of functions (f, f), where

There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be R. For example, (in particular in algebraic geometry), one could take this to be the space of complex numbers C equipped with the sheaf of holomorphic functions (thus arriving at the spaces of complex analytic geometry), or the sheaf of polynomials (thus arriving at the spaces of interest in complex algebraic geometry). In broader terms, this concept can be adapted for any suitable notion of a scheme (see topos theory). Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair (f, f), but these merely quantify the idea of local isomorphism rather than being central to the discussion (as in the case of charts and atlases). Third, the sheaf O is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals). Hence, it is a more primitive definition of the structure (see synthetic differential geometry).

A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.