  In this paper, we show that if $X$ is a smooth variety of general type of
dimension $m \geq 3$ for which the canonical map induces a triple cover onto
$Y$, where $Y$ is a projective bundle over $\mathbf P^1$ or onto a projective
space or onto a quadric hypersurface, embedded by a complete linear series,
then the general deformation of the canonical morphism of $X$ is again
canonical and induces a triple cover. The extremal case when $Y$ is embedded as
a variety of minimal degree is of interest, due to its appearance in numerous
situations. This is especially interesting as well, since it has no lower
dimensional analogues.
