In this talk, I will discuss some results concerning the geometry and topology of manifolds on which the first eigenvalue of the operator -γΔ + Ric is bounded below. Here, γ is a positive number, Δ is the Laplacian, and Ric denotes the pointwise lowest eigenvalue of the Ricci tensor.First, I will discuss some motivations and applications of such a study. Then, I will discuss spectral generalizations of the Bishop -- Gromov and Bonnet -- Myers theorems. If time allows, I will discuss a spectral generalization of the Cheeger -- Gromoll splitting theorem as well.This is joint work with M. Pozzetta and K. Xu.