Let $Z\subset P^N$ be a Fano manifold whose Picard group is generated by the hyperplane section class. Assume that $Z$ is covered by lines and $i\left(Z\right)\ge 3$. Let $？:X^Z\to Z$ be a double cover, branched along a smooth hypersurface section of degree $2m$,$1\le m\le i\left(Z\right)-2$. We describe the defining ideal of the variety of minimal rationaltangents at a general point. As an application, we show that if $Z\subset P^N$ is defined by quadratic equations and $2\le m\le i\left(Z\right)-2$, then the morphism $？$ satisfies the Cartan–Fubini type rigidity property.