We describe extensions of several key concepts of differential geometry to manifolds with density, including curvature, the Gauss-Bonnet theorem and formula, geodesics, and constant curvature surfaces.

Author Bio

Corwin, Hurder, Sesum and Xu worked together with Professor Frank Morgan over the summer of 2004 at Williams college as part of the SMALL REU program, sponsored partially by NSF grants. They had a chance to work with Hoffman while at a two week long conference in Paris./p>