  In this paper we study higher even Gaussian maps of the canonical bundle on
hyperelliptic curves and we determine their rank, giving explicit descriptions
of their kernels. Then we use this descriptions to investigate the
hyperelliptic Torelli map $j_h$ and its second fundamental form. We study
isotropic subspaces of the tangent space $T_{{\mathcal H}_g, [C]}$ to the
moduli space ${\mathcal H}_g$ of hyperelliptic curves of genus $g$ at a point
$[C]$, with respect to the second fundamental form $\rho_{HE}$ of $j_h$. In
particular, for any Weierstrass point $p \in C$, we construct a subspace $V_p$
of dimension $\lfloor\frac{g}{2} \rfloor$ of $T_{{\mathcal H}_g, [C]}$
generated by higher Schiffer variations at $p$, such that the only isotropic
tangent direction $\zeta \in V_p$ for the image of $\rho_{HE}$ is the standard
Schiffer variation $\xi_p$ at the Weierstrass point $p \in C$.
