Ruled Weingarten surfaces in Minkowski 3-space

Dillen, Franki; Kühnel, Wolfgang

doi:10.1007/s002290050142

Cited by 76 publications

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“…* E -m a i l :r c am i n o@ u g r . e s 1349 Brought to you by | MIT Libraries Authenticated Download Date | 5/11/18 2:09 PM The first part of this paper is motivated by the results from [2], where authors studied helicoidal surfaces generated by a straight-line, called helicoidal ruled surfaces. A particular case of such surfaces are the surfaces called right Lorentzian helicoids considered by Dillen and Kühnel, which appear when the axis L is timelike or spacelike and the curve γ is one of the coordinate axes of R 3 .…”

Section: T H a T I S T H E T H R E E -D Im E N S I O N A L R E A mentioning

confidence: 99%

“…All these three surfaces have zero mean curvature. When the axis L is lightlike, there are two known helicoidal surfaces generated by a straight-line called in the literature the Lie minimal surface (or the Cayley surface) and the parabolic null cylinder [2,13,16]. Both surfaces also have zero mean curvature.…”

Section: T H a T I S T H E T H R E E -D Im E N S I O N A L R E A mentioning

confidence: 99%

“…The axis is lightlike, L = (1 0 1) and = ( 0 ). As |α ( )| 2 = ( − ) 2 and ∈ L, α is the spacelike curve α( ) = ( 0 ) + ( − ) (0 1 0) + ( − )/2 2 (1 0 1). This curve lies in the plane − = − and from the Euclidean viewpoint, it is a parabola with the axis parallel to (1 0 1).…”

Section: Definition 24mentioning

confidence: 99%

“…By Proposition 2.2, we can suppose that the generating curve γ is contained in the plane = 0. If γ is not locally a graph on the -axis, then γ is the vertical line γ( ) = ( 0 ), whose corresponding helicoidal surface is the Lorentzian cylinder of equation 2 + 2 = 2 . This surface has constant mean curvature H = 1/(2 ).…”

Section: The Axis Is Timelikementioning

confidence: 99%

“…The leading coefficient is 4H 2 6 6 , a contradiction. If = 1, the expression (10) is a polynomial equation of degree 6, whose leading coefficient is 4H 2 1 − 2 Hence is a constant function and either H = 0 or ( ) = ± + 0 and H = 0. In the first case, the surface X ( ) = ( cos sin + 0 ) is the helicoid of first kind followed by a translation in the direction of the axis L.…”

Section: The Axis Is Timelikementioning

confidence: 99%

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Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

López

Demir

2014

Open Mathematics

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C e n t .E u r .J . M a t h . •1 2 ( 9 ) •2 0 1 4 •1 3 4 9 -1 3 6 1 DO I :1 0 . 2 4 7 8 / s 1 1 5 3 3 -0 1 4 -0 4 1 5 -0 Cen t ra l Eu ropean Jou rna l o f Ma the ma t ics H e l i c o i d a ls u r f a c e s i n M i n k ow s k is p a c ew i t hc o n s t a n t m e a nc u r v a t u r ea n dc o n s t a n tG a u s sc u r v a t u r e Resea rch A r t ic le R a f a e lL ó p e z 1 * ,E sm aD em i r 2 1 D e p a r t am e n t od eG e om e t r í ayT o p o l o g í a ,U n i v e r s i d a dd eG r a n a d a ,1 8 0 7 1G r a n a d a ,S p a i n 2 D e p a r tm e n to f M a t h em a t i c s ,F a c u l t yo fS c i e n c ea n dA r t s ,N e v s e h i rU n i v e r s i t y ,5 0 3 0 0N e v s e h i r ,T u r k e y Rece ived 11 Ap r i l 2013 ; accep ted 12 Dece mbe r 2013 Abs t rac t : W ec l a s s i f ya l lh e l i c o i d a ln o n -d e g e n e r a t es u r f a c e s i n M i n k ow s k is p a c e w i t hc o n s t a n t m e a nc u r v a t u r e w h o s e g e n e r a t i n gc u r v e i sa t h eg r a p ho fap o l y n om i a lo raL o r e n t z i a nc i r c l e . I n t h efi r s tc a s e ,w ep r o v e t h a t t h ed e g r e e o f t h ep o l y n om i a l i s0o r1a n d t h a t t h es u r f a c e i s r u l e d .I f t h eg e n e r a t i n gc u r v e i saL o r e n t z i a nc i r c l e ,w ep r o v e t h a t t h eo n l yp o s s i b i l i t y i s t h a t t h eax i s i ss p a c e l i k ea n d t h ec e n t e ro f t h ec i r c l e l i e so n t h ea x i s .

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Section: T H a T I S T H E T H R E E -D Im E N S I O N A L R E A mentioning

confidence: 99%

Section: T H a T I S T H E T H R E E -D Im E N S I O N A L R E A mentioning

confidence: 99%

Section: Definition 24mentioning

confidence: 99%

Section: The Axis Is Timelikementioning

confidence: 99%

Section: The Axis Is Timelikementioning

confidence: 99%

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Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

López

Demir

2014

Open Mathematics

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The 2‐ruled hypersurfaces in Minkowski 4‐space and their constructions via octonions

Ndiaye,

Özdemir

2023

Math Methods in App Sciences

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In this paper, we define three types of 2‐ruled hypersurfaces in the Minkowski 4‐space . We obtain Gaussian and mean curvatures of the 2‐ruled hypersurfaces of type‐1 and type‐2 and some characterizations about its minimality. We also deal with the first Laplace–Beltrami operators of these types of 2‐ruled hypersurfaces in . Moreover, the importance of this paper is the definition of these surfaces by using the octonions in . Thus, this is a new idea and makes the paper original. We give an example of 2‐ruled hypersurface constructed by octonion, and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one‐dimensional object embedded in the four‐dimensional Minkowski space . Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2‐ruled hypersurfaces in a four‐dimensional Minkowski space.

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