On the infinitesimal rigidity of homogeneous varieties

Let $X\subset P^N$ be a variety (respectively a patch of an analytic submanifold) and let $x\in X$ be a general point. We show that if the projective second fundamental form of $X$ at $x$ is isomorphic to the second fundamental form of a point of a Segre $P^n\times P^m$, $n,m\geq 2$, a Grassmaniann $G(2,n+2)$, $n\geq 4$, or the Cayley plane $OP^2$, then $X$ is the corresponding homogeneous variety (resp. a patch of the corresponding homogeneous variety). If the projective second fundamental form of $X$ at $x$ is isomorphic to the second fundamental form of a point of a Veronese $v_2(P^n)$ and the Fubini cubic form of $X$ at $x$ is zero, then $X=v_2(P^n)$ (resp. a patch of $v_2(P^n)$). All these results are valid in the real or complex analytic categories and locally in the $C^{\infty}$ category if one assumes the hypotheses hold in a neighborhood of any point $x$. As a byproduct, we show that the systems of quadrics $I_2(P^{m-1}\sqcup P^{n-1}), I_2(P^1\times P^{n-1})$ and $I_2(S_5)$ are stable in the sense that if $A_t\subset S^2T^*$ is an analytic family such that for $t\neq 0$, $A_t\simeq A$, then $A_0\simeq A$. We also make some observations related to the Fulton-Hansen connectedness theorem.