Ruled surfaces in 3-dimensional Riemannian manifolds

TL;DR

This paper investigates ruled surfaces in 3D Riemannian manifolds, deriving curvature expressions, introducing the Sannia frame, and characterizing striction curves to understand their geometric properties and existence conditions.

Contribution

It provides explicit formulas for curvatures, introduces the Sannia frame, and characterizes striction curves, advancing the understanding of ruled surfaces in Riemannian geometry.

Findings

Extrinsic curvature of ruled surfaces is non-positive.

Sannia frame allows a systematic study of ruled surfaces.

Striction curves are characterized as zeros of the Jacobi evolution function.

Abstract

In this work, ruled surfaces in 3-dimensional Riemannian manifolds are studied. We determine the expression for the extrinsic and sectional curvature of a parametrized ruled surface, where the former one is shown to be non-positive. We also quantify the set of ruling vector fields along a given base curve which allows to define a relevant reference frame that we refer to as Sannia frame. The fundamental theorem of existence and equivalence of Sannia-ruled surfaces in terms of a system of invariants is given. The second part of the article tackles the concept of the striction curve, which is proven to be the set of points where the so-called Jacobi evolution function vanishes on a ruled surface. This characterization of striction curves provides independent proof for their existence and uniqueness in space forms and disproves their existence or uniqueness in some other cases.