On Helicoidal Surfaces in a Conformally Flat 3-Space

Lee, Chul Woo; Lee, Jae Won; Yoon, Dae Won

doi:10.1007/s00009-017-0967-x

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…The manifold R 3 endowed with a metric as defined above will be denoted by E 3 F . Helicoidal minimal surfaces and helicoidal surfaces with prescribed extrinsic curvature were studied in E 3 F , for a special conformal factor [2], [11]. As this space is invariant under the actions of the rotational group around x 3 -axis, it is logical to consider rotational surfaces around x 3 -axis, because they are invariant under the action of the same group.…”

Section: Introductionmentioning

confidence: 99%

Complete surfaces with zero curvatures in conformally flat spaces

Araújo¹,

Corro²,

Pina³

et al. 2020

Publ. Math. Debrecen

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In this paper, we introduce a family of Riemannian manifolds E 3 F , which are Euclidean space R 3 endowed with conformally flat metrics. We characterize rotational surfaces with constant Gaussian and extrinsic curvatures in E 3 F . We present a particular space that is isometric to H 2 × S 1 , and, using a special parametrization, we construct a family of complete rotational surfaces with zero Gaussian and extrinsic curvatures in H 2 × S 1 . We have built a special space that is a warped product H 2 × f R, which is a complete space foliated by complete surfaces of constant Gaussian curvature −1; this shows that the hyperbolic space H 2 is isometrically immersed into the space H 2 × f R, and this space is isometric to neither H 3 nor H 2 × R, showing that in the ambient space, H 2 × f R Hilbert theorem does not hold.

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Section: Introductionmentioning

confidence: 99%

Complete surfaces with zero curvatures in conformally flat spaces

Araújo¹,

Corro²,

Pina³

et al. 2020

Publ. Math. Debrecen

View full text Add to dashboard Cite

show abstract

“…It is well known that a helicoidal surface is a generalization of a rotation surface. There are many studies about these surfaces under some given certain conditions [1][2][3][4][5][6][7][8][9][10][11][12]. Recently, the popular question has become whether a helicoidal surface can be constructed when its curvatures are prescribed.…”

Section: Introductionmentioning

confidence: 99%

“…This problem is extended to complete manifolds. Lee et al studied the helicoidal surfaces with a prescribed extrinsic curvature or mean curvature in a conformally flat 3-space [10]. It is well known that a metric on a complete manifold is conformal to the Euclidean metric.…”

Section: Introductionmentioning

confidence: 99%