Ruled surfaces in $3$-dimensional Riemannian manifolds

Marco Castrillón, M. Eugenia Rosado María, Alberto Soria

Published 2023-07-24Version 1

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 allow to define a relevant reference frame along it and that we refer to as \emph{Sannia}. 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 striction curve, which is proven to be the set of points where the so-called \emph{Jacobi evolution function} vanishes on a ruled surface. This provides independent proofs for their existence and uniqueness in space forms, and to disprove its existence or uniqueness in some other cases.