Characterization of manifolds of constant curvature by ruled surfaces

TL;DR

This paper studies ruled surfaces in 3D Riemannian manifolds, deriving fundamental forms and curvatures, and characterizes when such surfaces imply the manifold is a space form, with applications to hyperbolic and spherical products.

Contribution

It provides a model-independent proof of extrinsically flat ruled surfaces and characterizes manifolds based on the properties of these surfaces.

Findings

Extrinsically flat surfaces in space forms are ruled.

Necessary and sufficient conditions for ruled surfaces in generic 3D manifolds.

Existence of non-constant angle extrinsically flat surfaces in hyperbolic and spherical products.

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.