  Given a Riemannian manifold $N^n$ and ${\cal Z}\in \mathfrak{X}(N)$, an
isometric immersion $f\colon M^m\to N^n$ is said to have the \emph{constant
ratio property with respect to ${\cal Z}$} either if the tangent component
${\cal Z}^T_f$ of ${\cal Z}$ vanishes identically or if ${\cal Z}^T_f$ vanishes
nowhere and the ratio $\|{\cal Z}^\perp_f\|/\|{\cal Z}^T_f\|$ between the
lengths of the normal and tangent components of ${\cal Z}$ is constant along
$M^m$. It has the \emph{principal direction property with respect to ${\cal
Z}$} if ${\cal Z}^T_f$ is an eigenvector of all shape operators of $f$ at all
points of $M^m$. In this article we study isometric immersions $f\colon M^m\to
N^n$ of arbitrary codimension that have either the constant ratio or the
principal direction property with respect to distinguished vector fields ${\cal
Z}$ on space forms, product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$, where
$\Sf^n$ and $\Hy^n$ are the $n$-dimensional sphere and hyperbolic space,
respectively, and, more generally, on warped products $I\times_{\rho}\Q_\e^n$
of an open interval $I\subset \R$ and a space form $\Q_\e^n$. Starting from the
observation that these properties are invariant under conformal changes of the
ambient metric, we provide new characterization and classification results of
isometric immersions that satisfy either of those properties, or both of them
simultaneously, for several relevant instances of ${\cal Z}$ as well as simpler
descriptions and proofs of some known ones for particular cases of ${\cal Z}$
previously considered by many authors.
