Manifold structure of the set of all smooth functions between two smooth manifolds!

Question

In the studies of the calculus of variation, a map $f:M\to N$ said to be harmonic if it is a critical point of the Dirichlet energy function. i. e.

\begin{align} E:C^\infty(M,N)&\longrightarrow \Bbb R\\ f&\mapsto E(f) \end{align}

In general, a point $p$ in $M_1$ is a critical point of $f:M_1\to M_2$ if the differential $f_{*p}:T_pM_1\to T_{f(p)}M_2$ fails to be surjective. If $M_2=\Bbb R$ then $f$ has a critical point at $p \in M$ if and only if $(df)_p=0$.

How to show that $C^\infty(M,N)$ is an infinite dimensional smooth manifold and What is its tangent space at a point $f\in C^\infty(M,N)$?

Thanks.

Answer 1

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space at $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be constructed with the exponential map of $N$.