Compact Minimal Hypersurfaces with Index One in the Real Projective Space

Let M be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S(1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP = Sn+1(1)/{±}, obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface S n1(R1)×Sn2(R2) ⊂ S(1) obtained as the product of two spheres of dimensions n1, n2 and radius R1, R2, with n1 + n2 = n, R 1 + R 2 2 = 1 and n1R 2 2 = n2R 2 1.