Evolution equations of curvature tensors along the hyperbolic geometric flow

Details Evolution equations of curvature tensors along the hyperbolic geometric flow Evolution equations of curvature tensors along the hyperbolic geometric flow

2012-04-06

arxiv.org/abs/1204.1396v1

Mathematics

Differential Geometry

15 pages

Scientific paper

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

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