$U\ni 0$ be open in $X$, let $P:U\rightarrow Y$ with $P(0)=0$ and $L_0P:X\rightarrow Y$ surjective with finite-dimensional kernel $K$.

Then for some $U$, $P^{-1}(0)=\{u\in U\,:\, P(u)=0\}$ is a manifold of dimension $\dim K$. Moreover, if we write $X=K\oplus Z$, $P^{-1}(0)=\text{\emph{Graph}}\, G$ for some map $G$ from an open set in $K$ to $Z$ with $G(0)=0$.

This gives us a way to describe all perturbations of a given calibrated submanifold, as we now see in the special Lagrangian case, due to McLean [@McLean].

\[SLdef.thm\] Let $N$ be a compact special Lagrangian in a Calabi–Yau manifold $M$. Then the moduli space of deformations of $N$ is a smooth manifold of dimension $b^1(N)$.

One should compare this result to the deformation theory for complex submanifolds in Kähler manifolds. There, one does not get that the moduli space is a smooth manifold: in fact, it can be singular, and one has *obstructions* to deformations. It is somewhat remarkable that special Lagrangian calibrated geometry enjoys a much better deformation theory than this classical calibrated geometry. The deformation theory of embedded compact complex submanifolds in Calabi–Yau manifolds has recently been revisited using analytic techniques [@Moore.cpt].

The tubular neighbourhood theorem gives us a diffeomorphism $\exp:S\subseteq\nu(N)\rightarrow T\subseteq M$ 