Let X subset of P-N be a variety (respectively an open subset of an analytic submanifold) and let x is an element of X be a point where all integer valued differential invariants are locally constant. 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 x P-n, n, m greater than or equal to 2, a Grassmaniann G(2,n + 2), n greater than or equal to 4, or the Cayley plane OP2, then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P-2 x P-2 had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese upsilon(2)(P-n) and the Fubini cubic form of X at x is zero, then X = upsilon(2) (P-n) (resp. an open subset of upsilon(2)(P-n)). All these results are valid in the real or complex analytic categories and locally in the C-infinity 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(Pm-1 boolean OR Pn-1) subset of (SCm+n)-C-2, I-2(P-1 x Pn-1) subset of (SC2n)-C-2 and I-2(S-5) subset of (SC16)-C-2 are stable in the sense that if A(t) subset of (ST)-T-2* is an analytic family such that for t not equal 0, A(t) similar or equal to A, then A(0) similar or equal to A. We also make some observations related to the Fulton-Hansen connectedness theorem.
