  The commuting vector fields approach, devised for strichartz estimates in
[13], was developed for proving the local well-posedness in the Sobolev spaces
$H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in
${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they
obtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuum
Einstein equations, by taking advantage of the vanishing Ricci curvature. The
sharp, $H^{2+\epsilon}$, local well-posedness result for general quasilinear
wave equation was achieved by Smith and Tataru by constructing a parametrix
using wave packets. Using the vector fields approach, one has to face the major
hurdle caused by the Ricci tensor of the metric for the quasi-linear wave
equations. This posed a question that if the geometric approach can provide the
sharp result for the non-geometric equations. In this paper, based on geometric
normalization and new observations on the mass aspect function, we prove the
sharp local well-posedness of general quasilinear wave equation in ${\Bbb
R}^{1+3}$ by a vector field approach.
