The First Eigenvalue of $P$-manifolds

Antonio Ros gave a lower bound for the first eigenvalue ${\lambda }_1$ of $\mathrm{\Delta }$ of a $P$-manifold $(M,g)$ in terms of the lower bound on the Ricci curvature $Ri{c}_M$ and asked what happened when this lower bound was achieved. In this paper we look in to this question and show that there are strong implications on the geometry and topology of the underlying manifold. In particular we show that in case of spheres or real projective spaces we have isometry with the standard metric. In other cases, with some additional hypothesis, we again show isometry with standard models.