There exists a Riemannian metric on the real projective space such that the first eigenvalue coincides with that of its Riemannian universal cover, if the dimension is bigger than 2. For the proof, we deform the canonical metric on the real projective space. A similar result is obtained for lens spaces, as well as for closed Riemannian manifolds with Riemannian double covers. As a result, on a non-orientable closed manifold other than the real projective plane, there exists a Riemannian metric such that the first eigenvalue coincides with that of its Riemannian double cover.