  Inspired by a pioneer work of Andersson-Kapitanski \cite{AK}, we prove the
local well-posedness of the Cauchy problem of incompressible neo-Hookean
equations if the initial deformation and velocity belong to
$H^{s+1}(\mathbb{R}^n) \times H^{s}(\mathbb{R}^n), s>\frac{n+1}{2}$ ($n=2,3$).
Moreover, if the initial data is small, then we can lower the regularity to
$s>\frac{n}{2}$, where $\frac{n+2}{2}$ and $\frac{n}{2}$ is respectively a
scaling-invariant exponent for deformation and velocity in Sobolev spaces. Our
new observation relies on two folds: a reduction to a second-order
wave-elliptic system of deformation and velocity; and a "wave-map type" null
form intrinsic in this coupled system. In particular, the wave nature with
"wave-map type" null form allows us to prove a bilinear estimate of
Klainerman-Machedon type for nonlinear terms. So we can lower $\frac12$-order
regularity in 3D and $\frac34$-order regularity in 2D for well-posedness
compared with \cite{AK}.
