I am studying Ricci flow theory and the Ricci flow is not a parabolic equation. But there is some variant of it, called DeTurck-Ricci equation, that happens to be a parabolic PDE. So to argue existence in short time the books I am reading claim: "since the DeTurck-Ricci equation is parabolic, we can ensure existence for short time".

My question is: does the existence of solution is of classic solutions? Which theorem is this claimed? I could not find it in any PDE book I searched.

Thanks in advance