  We consider a geometric flow introduced by Gigli and Mantegazza which, in the
case of smooth compact manifolds with smooth metrics, is tangen- tial to the
Ricci flow almost-everywhere along geodesics. To study spaces with geometric
singularities, we consider this flow in the context of smooth manifolds with
rough metrics with sufficiently regular heat kernels. On an appropriate non-
singular open region, we provide a family of metric tensors evolving in time
and provide a regularity theory for this flow in terms of the regularity of the
heat kernel.
  When the rough metric induces a metric measure space satisfying a Riemannian
Curvature Dimension condition, we demonstrate that the distance induced by the
flow is identical to the evolving distance metric defined by Gigli and
Mantegazza on appropriate admissible points. Consequently, we demonstrate that
a smooth compact manifold with a finite number of geometric conical
singularities remains a smooth manifold with a smooth metric away from the cone
points for all future times. Moreover, we show that the distance induced by the
evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by
Gigli-Mantegazza.
