Geometry of submanifolds with respect to ambient vector fields

Abstract

Given a Riemannian manifold $N^n$ and ${\mathcal{Z}}\in {\mathfrak {X}}(N)$, an isometric immersion $f:M^m\rightarrow N^n$ is said to have the constant ratio property with respect to ${\mathcal{Z}}$ either if the tangent component ${\mathcal{Z}}^T_f$ of ${\mathcal{Z}}$ vanishes identically or if ${\mathcal{Z}}^T_f$ vanishes nowhere and the ratio $\Vert {\mathcal{Z}}^\perp _f\Vert /\Vert {\mathcal{Z}}^T_f\Vert$ between the lengths of the normal and tangent components of ${\mathcal{Z}}$ is constant along $M^m$. It has the principal direction property with respect to ${\mathcal{Z}}$ if ${\mathcal{Z}}^T_f$ is an eigenvector of all shape operators of f at all points of $M^m$. In this article, we study isometric immersions $f:M^m\rightarrow N^n$ of arbitrary codimension that have either the constant ratio or the principal direction property with respect to distinguished vector fields ${\mathcal{Z}}$ on space forms, product spaces ${\mathbb {S}}^n\times {\mathbb {R}}$ and ${\mathbb {H}}^n\times {\mathbb {R}}$, where ${\mathbb {S}}^n$ and ${\mathbb {H}}^n$ are the n-dimensional sphere and hyperbolic space, respectively, and, more generally, on warped products $I\times _{\rho }{\mathbb {Q}}_\epsilon ^n$ of an open interval $I\subset {\mathbb {R}}$ and a space form ${\mathbb {Q}}_\epsilon ^n$. Starting from the observation that these properties are invariant under conformal changes of the ambient metric, we provide new characterizations and classification results of isometric immersions that satisfy either of those properties, or both of them simultaneously, for several relevant instances of ${\mathcal{Z}}$ as well as simpler descriptions and proofs of some known ones for particular cases of ${\mathcal{Z}}$ previously considered by many authors. Our methods also allow us to classify Euclidean submanifolds with the property that the normal components of their position vector fields are parallel with respect to the normal connection, and to give alternative descriptions to those in Chen (J Geom 74(1–2): 61–77, 2002) of Euclidean submanifolds whose tangent or normal components of their position vector fields have constant length.