Rigidity of conformal submersions and quasi-Einstein manifolds

In this paper, we study two notions of rigidity, one of conformal submersions and the other of quasi Einstein manifolds, with an attempt to relate the two notions. Note that a smooth submersion between Riemannian manifolds is called conformal if it restricts to a conformal isometry on the horizontal distribution. A conformal submersion is said to be rigid if it reduces to a Riemannian submersion up to homothety. On the other hand, quasiEinstein manifolds are generalizations of Einstein manifolds that are of interest both in Riemannian geometry and theoretical physics. A Riemannian manifold $(M, g)$ is called quasi-Einstein if its Ricci tensor satisfies the identity: $R i c_g+ H e s s(f)-\frac{1}{m} d f \otimes d f=\lambda g$ for some $f \in C^{\infty}(M)$ and constants $\lambda \in \mathbb{R}$ and $0<m \leq \infty$. A quasi-Einstein manifold is said to be rigid if it reduces to an Einstein manifold. In this paper, we employ certain techniques involving conformal submersions to establish rigidity results for a class of closed quasiEinstein manifolds with $\lambda>0$. In particular, we study curvature conditions that force conformal submersions to be rigid, also leading to the rigidity of a related class of quasi-Einstein manifolds.