Eigenvalues and lambda constants on Riemannian submersions

Given a Riemannian submersion, we study the relation between lambda constants introduced by G. Perelman on the base manifold and the total space of a Riemannian submersion. We also discuss the relationship between the first eigenvalues of Laplacians on the base manifold and that of the total space. The quantities on warped products are discussed in detail.