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.



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


