Compact minimal hypersurfaces with index one in the real projective space

Let $${M}^{n} $$ be a compact (two-sided) minimal hypersurface in a Riemannian manifold $$ \overline{M}^{n+1} .$$ It is a simple fact that if $$ \overline{M}$$ has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If $$ \overline{M}=S{n+1} $$ 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 $$ {\overline{M}}$$ is the real projective space $${P^{n+1}}={S^{n+1}}/\{\pm\},$$ 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 $${S^n1}({R_1})\times{S^n2}({R_2})\subset{S^{n+1}}$$ obtained as the product of two spheres of dimensions $${n_1}, {n_2}$$ and radius $${R_1}, {R_2}$$ , with $${n_1}+{n_2}=n,\,{R_1^2}+{R_2^2}=1\,{\rm and}\,{R_2^2}={n_2}{R_1^2}$$