Knowlpedia

Definition

Sheaf of smooth functions

The sheaf assigning to each open subset of a smooth manifold its algebra of smooth real-valued functions.

Definition

Let MM be a . Its sheaf of smooth functions, denoted CM∞C^\infty_M, assigns to every open set U⊆MU\subseteq M the commutative unital real algebra

CM∞(U)={f:U→R∣f is smooth},C^\infty_M(U)=\{f:U\to\mathbb R\mid f\text{ is smooth}\},

with restriction maps given by restricting functions. Smooth functions that agree on overlaps glue uniquely, and smoothness can be checked locally in charts, so CM∞C^\infty_M is a of R\mathbb R-algebras. The pair (M,CM∞)(M,C^\infty_M) is the smooth manifold regarded as a .

Stalks and local structure

The CM,p∞C^\infty_{M,p} consists of germs of smooth real-valued functions near pp. It is a : its unique contains exactly the germs that vanish at pp, and evaluation at pp identifies the with R\mathbb R. Unlike the holomorphic case, a smooth germ is not determined by its Taylor series; nonzero flat germs have every derivative equal to zero at the base point.

Pullback and geometric meaning

Every F:M→NF:M\to N pulls functions back by composition and therefore induces local homomorphisms

CN,F(p)∞⟶CM,p∞.C^\infty_{N,F(p)}\longrightarrow C^\infty_{M,p}.

The sheaf records the smooth structure algebraically: act as local derivations of it, differential forms can be organized as modules over it, and let local smooth constructions be patched globally. This algebraic viewpoint is developed in Nestruev, Chapters 1–5.

Conventions and scope
References
  1. Jet Nestruev, Smooth Manifolds and Observables, 2nd ed., Graduate Texts in Mathematics 220, Springer, 2020. Publisher record. Relevant: Chapters 1–5, smooth functions and the algebraic description of manifolds.
  2. Ieke Moerdijk and Janez Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, 2003. Publisher record. Relevant: prerequisites and Appendix A, smooth manifolds, sheaves, and partitions of unity.