2019/09/06 by Manfio, Fernando, Tojeiro, Ruy, Van der Veken, Joeri
#53B25 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.02892
Given a Riemannian manifold Nn and \cal Z∈ \mathfrakX(N), an isometric immersion f\colon Mm→ Nn is said to have the \emphconstant ratio property with respect to \cal Z either if the tangent component \cal ZTf of \cal Z vanishes identically or if \cal ZTf vanishes nowhere and the ratio ‖\cal Z^⊥f‖/‖\cal ZTf‖ between the lengths of the normal and tangent components of \cal Z is constant along Mm. It has the \emphprincipal direction property with respect to \cal Z if \cal ZTf is an eigenvector of all shape operators of f at all points of Mm. In this article we study isometric immersions f\colon Mm→ Nn 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 \Sfn× \R and \Hyn× \R, where \Sfn and \Hyn are the n-dimensional sphere and hyperbolic space, respectively, and, more generally, on warped products I×ρ\Q_\en of an open interval I⊂ \R and a space form \Q_\en. 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.