Space of smooth maps - Imported Knowlpedia Content

Definition

For finite-dimensional smooth manifolds $M$ and $N$ , the smooth maps from $M$ to $N$ form the space of smooth maps

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

As a set, it is the morphism set $\operatorname{Hom}_{\mathbf{Man}}(M,N)$ in the smooth-manifold category. When $M$ is compact, the standard mapping-space convention equips it with the compact-open $C^\infty$ topology and an infinite-dimensional smooth-manifold structure; near $f$ , charts are modeled on smooth sections of $f^*TN$ . For noncompact $M$ , the phrase “mapping space” does not determine a unique topology or calculus without further conventions.

Local model and tangent space

Choose a local addition on $N$ , such as one obtained from a Riemannian exponential map. It identifies maps near $f$ with sections near the zero section of the pullback tangent bundle $f^*TN$ . For compact $M$ , the tangent space at $f$ is therefore

T_fC^\infty(M,N)\cong\Gamma(M,f^*TN).

If $N=\mathbb R^q$ , the mapping space is the Fréchet space $C^\infty(M,\mathbb R^q)$ .

Evaluation and composition

In the convenient smooth structure, evaluation

\operatorname{ev}:C^\infty(M,N)\times M\to N,\qquad(f,x)\mapsto f(x),

is smooth, and composition of smooth maps is smooth as a map between the corresponding mapping spaces. This cartesian-closed behavior is one reason to retain the infinite-dimensional smooth structure rather than only the underlying set Kriegl and Michor, Chapter IX.

Topology and scope

For noncompact source manifolds, compact-open $C^\infty$ , weak Whitney, strong Whitney, and convenient mapping-space constructions can lead to different topological or manifold structures. Claims about continuity, tangent spaces, or smooth composition must therefore name the framework and hypotheses. The bare notation $C^\infty(M,N)$ safely denotes only the set of smooth maps.

References

Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs 53, AMS, 1997. AMS record. Relevant: Chapter IX, manifolds of mappings.
Peter W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series 3, Shiva Publishing, 1980. Author record. Relevant: mapping-space topologies and smooth structures.