Curvature and geometric equations in smooth metric measure spaces

Abstract

A semi-Riemannian manifold is endowed with a density function, modifying the Riemannian volume element and giving rise to a smooth metric measure space. These spaces appear naturally in many topics in Mathematics, and their study combines ideas from Geometry, Topology and Analysis. In this thesis, we tackle the study of geometric equations in smooth metric measure spaces. The results presented can be broadly divided into two lines. The first one concerns a natural generalization of Einstein manifolds for Riemannian smooth metric measure spaces called weighted Einstein manifolds. We classify manifolds with this property which additionally satisfy a harmonicity condition on a weighted analogue of the Weyl tensor. We also translate a classical problem of Riemannian geometry to this setting and classify weighted Einstein manifolds admitting another such structure in their conformal class.