Nonsmooth differential geometry - An approach tailored for spaces with   Ricci curvature bounded from below
  We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature. 
Nicola Gigli
math.DG math.MG