Resumen de Curvature and geometric equations in smooth metric measure spaces
Diego Mojón Álvarez
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. The second line concerns the derivation of vacuum weighted Einstein field equations for Lorentzian smooth metric measure spaces. We present both a variational approach and an alternative perspective based on the characterizing properties of the usual Einstein tensor. We classify solutions under conditions on the density and the geometry of the underlying manifold. Among others, we study solutions whose density has lightlike gradient, and also general solutions which have harmonic curvature. Special families of Kundt spacetimes such as Brinkmann waves and pp-waves play a distinguished role as solutions with specific features.
