Compact minimal hypersurfaces with index one in the real projective space

Let M-n in be a compact (two-sided) minimal hypersurface in a Riemannian manifold (M) over bar(n+l). It is a simple fact that if (M) over bar has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If (M) over bar = Sn+l is the unit sphere and L has index one, then it is known that nl must Le a totally geodesic equator. We prove that if (M) over bar is the real projective space Pn+1 = Sn+1/{+/-}, obtained as a metric quotient of the unit sphere, and the Jacobi operator of nd has index one, then ni is either a totally geodesic sphere or the quotient to the projective space of the hypersurface S-n1 (R-1) x S-n2(R-2) subset of Sn+1 obtained as the product of two spheres of dimensions n(1),n(2) and radius R-1, R-2, with n(1) + n(2) = n, R-1(2) + R-2(2) = 1 and n(1)R(2)(2) = n(2)R(1)(2).