Second fundamental form and higher Gaussian maps

In this paper we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus $g\ge 4$ and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map ${\mu }_{6g-6}$ is injective, hence all even Gaussian maps ${\mu }_{2k}$ are identically zero for all $k>3g-3$. We also give an estimate for the rank of ${\mu }_{2k}$ for $g-1\le k\le 3g-3.$