Ruled surfaces in 3-dimensional Riemannian manifolds

Marco Castrillón, María Eugenia Rosado, Alberto Soria
Mathematics, Differential Geometry, Differential Geometry (math.DG)
2023-07-23 16:00:00
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.
PDF: Ruled surfaces in 3-dimensional Riemannian manifolds.pdf
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