Characterization of manifolds of constant curvature by ruled surfaces

Abstract

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifold has an extrinsically flat surface tangent to any 2d plane and if they are all ruled surfaces, then the manifold is a space form. As an application, we prove that there must exist extrinsically flat surfaces in the Riemannian product of the hyperbolic plane, or sphere, with the reals, and that do not make a constant angle with the real direction.