  Let $\phi: X \to \mathbb P^n$ be a double cover branched along a smooth
hypersurface of degree $2m, 2 \leq m \leq n-1$. We study the varieties of
minimal rational tangents $\mathcal C_x \subset \mathbb P T_x(X)$ at a general
point $x$ of $X$. We describe the homogeneous ideal of $\mathcal C_x$ and show
that the projective isomorphism type of $\mathcal C_x$ varies in a maximal way
as $x$ varies over general points of $X$. Our description of the ideal of
$\mathbb C_x$ implies a certain rigidity property of the covering morphism
$\phi$. As an application of this rigidity, we show that any finite morphism
between such double covers with $m=n-1$ must be an isomorphism. We also prove
that Liouville-type extension property holds with respect to minimal rational
curves on $X$.
