  In this paper we give an explicit bound of $\Delta_{g(t)}u(t)$ and the local
curvature estimates for the Ricci-harmonic flow under the condition that the
Ricci curvature is bounded along the flow. In the second part these local
curvature estimates are extended to a class of generalized Ricci flow,
introduced by the author \cite{LY1}, whose stable points give Ricci-flat
metrics on a complete manifold, and which is very close to the $(K, N)$-super
Ricci flow recently defined by Xiangdong Li and Songzi Li \cite{LL2014}. Next
we propose a conjecture for Einstein's scalar field equations motivated by a
result in the first part and the bounded $L^{2}$-curvature conjecture recently
solved by Klainerman, Rodnianski and Szeftel \cite{KRS2015}. In the last two
parts of this paper, we discuss two notions of "Riemann curvature tensor" in
the sense of Wylie-Yeroshkin \cite{KW2017, KWY2017, Wylie2015, WY2016},
respectively, and Li \cite{LY3}, whose "Ricci curvature" both give the standard
Bakey-\'Emery Ricci curvature \cite{BE1985}, and the forward and backward
uniqueness for the Ricci-harmonic flow.

Differential Geometry