The Levi-Civity connection described in https://en.wikipedia.org/wiki/Levi-Civita_connection specifically for torsion-free. The question is: do we have connection on the tangent bundle which takes into account torsion as well. The inclusion of torsion may be in terms of a parameter. I am looking for some reference to understand what is involve to do so.
Motivation: I am looking at Ricci flow. The project I have seems to require a solution of Ricci flow equations. The document in the link https://web.math.princeton.edu/~nsher/ricciflow.pdf explains quite a bit to derive the Ricci flow equation using Levi-Civity connection. But the project that I have, requires torsion to take into account.