Knowlpedia

Definition

Algebra of smooth functions

The unital commutative algebra of scalar-valued smooth functions on a smooth manifold under pointwise operations.

Definition

Let MM be a . The algebra of smooth functions on MM is

C∞(M)={f:M→R∣f is smooth}.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),(fg)(p)=f(p)g(p),(af)(p)=af(p).(f+g)(p)=f(p)+g(p),\qquad (fg)(p)=f(p)g(p),\qquad (af)(p)=a f(p).

With these operations, C∞(M)C^\infty(M) is a commutative unital R\mathbb R-algebra whose multiplicative identity is the constant function 11. For complex-valued functions the same construction gives the commutative unital C\mathbb C-algebra C∞(M,C)C^\infty(M,\mathbb C).

Geometric information in the algebra

Evaluation at p∈Mp\in M, f↦f(p)f\mapsto f(p), is a unital C∞(M)→RC^\infty(M)\to\mathbb R. Smooth functions separate distinct points and supply local coordinates and . Under the usual Hausdorff and second-countability hypotheses, this algebra therefore retains substantial information about the manifold; the algebraic viewpoint is developed systematically in Nestruev, chapters “Algebras and Points” and “Smooth Manifolds”.

Pullback and modules

A F:M→NF:M\to N induces a unital algebra homomorphism

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

Thus smooth manifolds map contravariantly to their smooth-function algebras. Spaces of smooth sections of over MM are naturally C∞(M)C^\infty(M)-modules, with multiplication defined pointwise.

Conventions and examples

The notation C∞(M)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=RnM=\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≠10\ne1.

References
  1. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: chapters on smooth manifolds and smooth maps.
  2. Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: “Algebras and Points,” “Smooth Manifolds,” and “Smooth Maps.”