Isometric Immersions with Geodesic Normal Sections in Semi-Riemannian Geometry

Abstract

We study an isometric immersion $f:M \to \bar{M}$ with geodesic normal sections, where $\bar{M}$ is a semi-Riemannian space form. In Riemannian geometry, it is known that $f$ is helical, in particular, all geodesics of $M$ have the same proper order in $\bar{M}$. However this does not hold in general, when $\bar{M}$ is indefinite semi-Riemannian. We give sufficient conditions for an isometric immersion with geodesic normal sections to be helical.