On the geometry of Riemannian manifolds with density

We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection motivates new versions of the volume and Laplacian comparison theorems that are valid for the 1-Bakry-Emery Ricci tensor, a weaker assumption than has previously been considered in the literature. As applications we prove new generalizations of Myers' theorem and Cheng's diameter rigidity result. We also investigate the holonomy groups of the weighted connection. We show that they are more general than the Riemannian holonomy, but also exhibit some of the same structure. For example, we obtain a generalization of the de Rham splitting theorem as well as new rigidity phenomena for parallel vector fields. A general feature of all of our rigidity results is that warped or twisted product splittings are characterized, as opposed to the usual isometric products.