Algebra of smooth functions

Let $M$ be a smooth manifold. The algebra of smooth functions on $M$ is

C^\infty(M)=\{f:M\to\mathbb R\mid f\text{ is smooth}\}.

Addition, multiplication, and scalar multiplication are defined pointwise:

(f+g)(p)=f(p)+g(p),\qquad (fg)(p)=f(p)g(p),\qquad (af)(p)=a f(p).

With these operations, $C^\infty(M)$ is a commutative unital $\mathbb R$ -algebra whose multiplicative identity is the constant function $1$ . For complex-valued functions the same construction gives the commutative unital $\mathbb C$ -algebra $C^\infty(M,\mathbb C)$ .

Canonical smooth functional calculus

The real algebra $C^\infty(M)$ carries more than its polynomial operations. Every smooth map $g:\mathbb R^n\to\mathbb R$ defines

\Phi_g(f_1,\ldots,f_n)(p) =g(f_1(p),\ldots,f_n(p)).

These operations make $C^\infty(M)$ a canonical $C^\infty$ -ring. In particular, one may apply $\sin$ , $\exp$ , and arbitrary multivariable smooth functions to elements. Forgetting these operations leaves the underlying commutative $\mathbb R$ -algebra.

Geometric information in the algebra

Evaluation at $p\in M$ , $f\mapsto f(p)$ , is a unital algebra homomorphism $C^\infty(M)\to\mathbb R$ . Smooth functions separate distinct points and supply local coordinates and bump functions. Under the usual Hausdorff and second-countability hypotheses, this algebra therefore retains substantial information about the manifold.

Pullback and modules

A smooth map $F:M\to N$ induces a unital algebra homomorphism

F^*:C^\infty(N)\to C^\infty(M),\qquad F^*h=h\circ F.

It preserves every operation $\Phi_g$ , so it is a $C^\infty$ -ring morphism. Thus smooth manifolds map contravariantly to their smooth-function algebras. For finite-dimensional Hausdorff second-countable manifolds, every $C^\infty$ -ring morphism in the opposite direction arises uniquely this way. Spaces of smooth sections of vector bundles over $M$ are naturally $C^\infty(M)$ -modules, with multiplication defined pointwise.

Conventions and examples

The notation $C^\infty(M)$ normally means real-valued functions unless the scalar field is stated. It includes all smooth functions, not only compactly supported ones. For $M=\mathbb R^n$ , it is the familiar algebra of functions with continuous derivatives of every order. For the empty manifold it is the zero ring, so some authors exclude that case when requiring a unital algebra with $0\ne1$ .

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: chapters on smooth manifolds and smooth maps.
Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: “Algebras and Points,” “Smooth Manifolds,” and “Smooth Maps.”
Ieke Moerdijk and Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, 1991. DOI record. Relevant: Chapter I, $C^\infty$ -rings.