An approach to energy and elastic for curves with extended Darboux frame in Minkowski space

It is known that a Bishop frame of a curve is one of the effective alternative approach in the differential geometry. Recently, several important works have been done about the Bishop frames. The aim of our paper is to investigate the N-Bishop frame for timelike curves in Minkowski space. We define the N-Bishop frame for the timelike curve in Minkowski space. Then, we consider some properties of the frame. Moreover, we describe the N-Bishop Darboux frame for the first time. Additionally, we compute some geometrical characterizations for the N-Bishop Darboux axis and momentum rotation vector.

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Isophote curves on spacelike surfaces in Lorentz–Minkowski space

Asian-European Journal of Mathematics ◽

10.1142/s1793557121501801 ◽

2021 ◽

pp. 2150180

Author(s):

Fatih Doğan ◽

Yusuf Yaylı

Keyword(s):

Minkowski Space ◽

Spacelike Surface ◽

Constant Angle ◽

Asymptotic Curves ◽

Lines Of Curvature ◽

Darboux Frame ◽

Curves On Surfaces ◽

Normal Vectors ◽

Spacelike Surfaces

An isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz–Minkowski space [Formula: see text] and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give some relations between isophote curves and special curves on surfaces such as geodesic curves, asymptotic curves or lines of curvature.

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N-Bishop Darboux vector of the spacelike curve with spacelike binormal

Thermal Science ◽

10.2298/tsci181112048k ◽

2019 ◽

Vol 23 (Suppl. 1) ◽

pp. 353-360

Author(s):

Hatice Kusak-Samanci ◽

Huseyin Kocayigit

Keyword(s):

Minkowski Space ◽

Rotation Vector ◽

Spacelike Curve ◽

Geometrical Properties ◽

Darboux Vector ◽

Bishop Frame ◽

Darboux Frame ◽

First Time

In this article, the N-Bishop frame in Minkowski space is investigated for spacelike curves with a spacelike binormial. Some features of the normal expansion are proven via for the spacelike curve. Then, a new Darboux frame called by the N-Bishop Darboux frame is introduced at first time. Furthermore, some geometrical properties of the N-Bishop Darboux frame are proven. As a result, the N-Bishop Darboux axis and momentum rotation vector are calculated.

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Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

Symmetry ◽

10.3390/sym13112185 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2185

Author(s):

Fatma Bulut

Keyword(s):

Differential Geometry ◽

Minkowski Space ◽

Symmetric Matrix ◽

Space Curve ◽

Frame Field ◽

Dimensional Minkowski Space ◽

Darboux Frame ◽

Surface Curve ◽

Area Of Study

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

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Ruled Surfaces with Constant Slope Ruling According to Darboux Frame in Minkowski Space

International Journal of Analysis and Applications ◽

10.28924/2291-8639-18-2020-900 ◽

2020 ◽

Keyword(s):

Minkowski Space ◽

Ruled Surfaces ◽

Darboux Frame ◽

Constant Slope

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Extended Darboux frame field in Minkowski space-time $mathbb{E}^4_1$

Malaya Journal of Matematik ◽

10.26637/mjm0603/0002 ◽

2018 ◽

Vol 06 (03) ◽

pp. 473-477 ◽

Cited By ~ 1

Author(s):

Bahar Uyar Düldül

Keyword(s):

Minkowski Space ◽

Space Time ◽

Frame Field ◽

Minkowski Space Time ◽

Darboux Frame

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On k-type spacelike slant helices lying on lightlike surfaces

Filomat ◽

10.2298/fil1909781o ◽

2019 ◽

Vol 33 (9) ◽

pp. 2781-2796

Author(s):

Ufuk Öztürk ◽

Emilija Nesovic ◽

Öztürk Koç

Keyword(s):

Minkowski Space ◽

Sufficient Conditions ◽

Geodesic Curvature ◽

Normal Curvature ◽

Spacelike Curve ◽

Bishop Frame ◽

Darboux Frame

In this paper, we define k-type spacelike slant helices lying on a lightlike surface in Minkowski space E31 according to their Darboux frame for k ? {0,1,2}. We obtain the necessary and the sufficient conditions for spacelike curves with non-null and null principal normal lying on lightlike surface to be the k-type spacelike slant helices in terms of their geodesic curvature, normal curvature and geodesic torsion. Additionally, we determine their axes and show that the Darboux frame of a spacelike curve lying on a lightlike surface coincides with its Bishop frame if and only if it has zero geodesic torsion. Finally, we give some examples.

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