Inflexible $CR$ submanifolds

In this paper we introduce the concept of inflexible $CR$ submanifolds. These are $CR$ submanifolds of some complex Euclidean space such that any compactly supported $CR$ deformation is again globally $CR$ embeddable into some complex Euclidean space. Our main result is that any $2$-pseudoconcave quadratic $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$ is inflexible.