Characterization of manifolds of constant curvature by spherical curves

Silva, Luiz C. B. da; Silva, José D. da

doi:10.1007/s10231-019-00874-5

Cited by 7 publications

(5 citation statements)

References 15 publications

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“…Further, we characterized space forms as the only manifolds with the properties that there exist an extrinsically flat surface tangent to any 2-plane and that these surfaces are all ruled. This result is part of our ongoing effort to find characteristic properties of space forms [21]. As an application of our results, we proved that there must exist extrinsically flat surfaces in the product manifolds H 2 × R and S 2 × R that do not make a constant angle with the real direction.…”

Section: Discussionsupporting

confidence: 62%

“…Finally, comparison with Eq. (21) shows that ψ(u, w) = λ u (β(v − w)), from which we conclude that Σ 2 is a helicoid.…”

Section: Ruled Minimal Surfacesmentioning

confidence: 52%

“…Namely, Eq. (53) which is crucial in our proof plays a role similar to that of the umbilicity condition ∇ X Y = −λX in [21]. This similarity may be interpreted in terms of rotation minimizing vector fields.…”

Section: Discussionmentioning

confidence: 69%

“…It is known that in Euclidean [23] and hyperbolic [12] spaces there is a close relationship between the notion of rotation minimizing frames and developable surfaces. (A vector field Z normal to a curve α : I → M 3 is said to minimize rotation along α if ∇ α Z and α are parallel [3,21,11].) The same relation is also valid in S 3 (r) and it follows as a corollary of Proposition 3: Proposition 4 The extrinsic curvature of a ruled surface vanishes if and only if α , Z, and ∇ α Z are linearly dependent.…”

Section: Remarkmentioning

confidence: 88%

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Characterization of manifolds of constant curvature by ruled surfaces

da Silva,

da Silva

2021

Preprint

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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 allows us to identify the necessary and sufficient condition 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 does not make a constant angle with the real direction.

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Section: Discussionsupporting

confidence: 62%

“…Finally, comparison with Eq. (21) shows that ψ(u, w) = λ u (β(v − w)), from which we conclude that Σ 2 is a helicoid.…”

Section: Ruled Minimal Surfacesmentioning

confidence: 52%

Section: Discussionmentioning

confidence: 69%

Section: Remarkmentioning

confidence: 88%

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Characterization of manifolds of constant curvature by ruled surfaces

da Silva,

da Silva

2021

Preprint

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“…Theorem 1.1 and the first part of Theorem 1.2 have been generalized to three-dimensional orientable Riemannian manifolds of constant curvature [ 8 ]; see also [ 2 , 15 ] for related results. In the present note we shall see that, under suitable assumptions, both theorems remain valid when is replaced by an arbitrary Riemannian manifold , provided one restricts the attention to three-dimensional curves; roughly speaking, a curve in M is three-dimensional if it has one curvature and one “torsion”, all other curvature functions being zero.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Total torsion of three-dimensional lines of curvature

Raffaelli

2023

Geom Dedicata

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A curve $$\gamma $$ γ in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $$\gamma $$ γ lies on an oriented hypersurface S of M, we say that $$\gamma $$ γ is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that $$\gamma $$ γ is three-dimensional and closed. We show that if $$\gamma $$ γ is a well-positioned line of curvature of S, then its total torsion is an integer multiple of $$2\pi $$ 2 π ; and that, conversely, if the total torsion of $$\gamma $$ γ is an integer multiple of $$2\pi $$ 2 π , then there exists an oriented hypersurface of M in which $$\gamma $$ γ is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of $$\gamma $$ γ vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.

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