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We consider the hyperbolic geometric flow $\frac{\partial^2}{\partialt^2}g(t)=-2Ric_{g(t)}$ introduced by Kong and Liu [KL]. When the Riemannianmetric evolve, then so does its curvature. Using the techniques and ideas ofS.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connectionand the curvature tensors along the hyperbolic geometric flow. The method andresults are computed and written in global tensor form, different from thelocal normal coordinate method in [DKL1]. In addition, we further show that anysolution to the hyperbolic geometric flow that develops a singularity in finitetime has unbounded Ricci curvature.

Evolution equations of curvature tensors along the hyperbolic geometric flow