Author:
López Marco Castrillón,Rosado M. Eugenia,Soria Alberto
Abstract
AbstractIn this work, ruled surfaces in 3-dimensional Riemannian manifolds are studied. We determine the expressions for the extrinsic and sectional curvatures 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 us 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 characterisation of striction curves provides independent proof for their existence and uniqueness in space forms and disproves their existence or uniqueness in some other cases.
Funder
Universidad Complutense de Madrid
Publisher
Springer Science and Business Media LLC
Reference23 articles.
1. Albujer, A.L., dos Santos, F.R.: On the geometry of non-degenerate surfaces in Lorentzian homogeneous $$3 $$-manifolds. arXiv preprint arXiv:2108.06823 (2021)
2. Castrillón López, M. , Fernández Mateos, V., Muñoz Masqué, J.: The equivalence problem of curves in a Riemannian manifold. Ann. Mat. Pura Appl. (1923) 194(2), 343–367 (2015)
3. Choi, S.M.: On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space. Tsukuba J. Math. 19(2), 285–304 (1995)
4. da Silva, L.C.B., da Silva, J.D.: Characterization of manifolds of constant curvature by ruled surfaces. São Paulo J. Math. Sci. 16(2), 1138–1162 (2022)
5. Dillen, F., Kühnel, W.: Ruled Weingarten surfaces in Minkowski $$3$$-space. Manuscr. Math. 98, 307–320 (1999)