• 제목/요약/키워드: timelike vector

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A GEOMETRIC APPROACH TO TIMELIKE FLOWS IN TERMS OF ANHOLONOMIC COORDINATES

  • Yavuz, Ayse;Erdogdu, Melek
    • 호남수학학술지
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    • 제44권2호
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    • pp.259-270
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    • 2022
  • This paper is devoted to the geometry of vector fields and timelike flows in terms of anholonomic coordinates in three dimensional Lorentzian space. We discuss eight parameters which are related by three partial differential equations. Then, it is seen that the curl of tangent vector field does not include any component in the direction of principal normal vector field. This implies the existence of a surface which contains both s - lines and b - lines. Moreover, we examine a normal congruence of timelike surfaces containing the s - lines and b - lines. Considering the compatibility conditions, we obtain the Gauss-Mainardi-Codazzi equations for this normal congruence of timelike surfaces in the case of the abnormality of normal vector field is zero. Intrinsic geometric properties of these normal congruence of timelike surfaces are obtained. We have dealt with important results on these geometric properties.

ON TIMELIKE BERTRAND CURVES IN MINKOWSKI 3-SPACE

  • Ucum, Ali;Ilarslan, Kazim
    • 호남수학학술지
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    • 제38권3호
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    • pp.467-477
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    • 2016
  • In this paper, we study the timelike Bertrand curves in Minkowski 3-space. Since the principal normal vector of a timelike curve is spacelike, the Bertrand mate curve of this curve can be a timelike curve, a spacelike curve with spacelike principal normal or a Cartan null curve, respectively. Thus, by considering these three cases, we get the necessary and sufficient conditions for a timelike curve to be a Bertrand curve. Also we give the related examples.

A NEW MODELLING OF TIMELIKE Q-HELICES

  • Yasin Unluturk ;Cumali Ekici;Dogan Unal
    • 호남수학학술지
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    • 제45권2호
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    • pp.231-247
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    • 2023
  • In this study, we mean that timelike q-helices are curves whose q-frame fields make a constant angle with a non-zero fixed axis. We present the necessary and sufficient conditions for timelike curves via the q-frame to be q-helices in Lorentz-Minkowski 3-space. Then we find some results of the relations between q-helices and Darboux q-helices. Furthermore, we portray Darboux q-helices as special curves whose Darboux vector makes a constant angle with a non-zero fixed axis by choosing the curve as one of the types of q-helices, and also the general case.

ON H2-PROPER TIMELIKE HYPERSURFACES IN LORENTZ 4-SPACE FORMS

  • Firooz Pashaie
    • 대한수학회논문집
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    • 제39권3호
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    • pp.739-756
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    • 2024
  • The ordinary mean curvature vector field 𝗛 on a submanifold M of a space form is said to be proper if it satisfies equality Δ𝗛 = a𝗛 for a constant real number a. It is proven that every hypersurface of an Riemannian space form with proper mean curvature vector field has constant mean curvature. In this manuscript, we study the Lorentzian hypersurfaces with proper second mean curvature vector field of four dimensional Lorentzian space forms. We show that the scalar curvature of such a hypersurface has to be constant. In addition, as a classification result, we show that each Lorentzian hypersurface of a Lorentzian 4-space form with proper second mean curvature vector field is C-biharmonic, C-1-type or C-null-2-type. Also, we prove that every 𝗛2-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.

VOLUME PROBLEMS ON LORENTZIAN MANIFOLDS

  • Kim, Seon-Bu
    • 대한수학회논문집
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    • 제10권1호
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    • pp.163-173
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    • 1995
  • Inspired in [2,9,10,17], pp. E. Ehrlich and S. B. Kim in [4] cultivated the Riccati equation related to the Raychaudhuri equation of General Relativity for the stable Jacobi tensor along the geodesics to extend the Hawking-Penrose conjugacy theorem to $$ f(t) = Ric(c(t)',c'(t)) + tr(\sigma(A)^2) $$ where $\sigma(A)$ is the shear tensor of A for the stable Jacobi tensor A with $A(t_0) = Id$ along the complete Riemannian or complete nonspacelike geodesics c.

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POSITION VECTORS OF A SPACELIKE W-CURVE IN MINKOWSKI SPACE 𝔼13

  • Ilarslan, Kazim;Boyacioglu, Ozgur
    • 대한수학회보
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    • 제44권3호
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    • pp.429-438
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    • 2007
  • In this paper, we study the position vectors of a spacelike W-curve (or a helix), i.e., curve with constant curvatures, with spacelike, timelike and null principal normal in the Minkowski 3-space $\mathbb{E}_1^3$. We give some characterizations for spacelike W - curves whose image lies on the pseudohyperbolical space $\mathbb{H}_0^2$ and Lorentzian sphere $\mathbb{S}_1^2$ by using the positions vectors of the curve.

BJÖRLING FORMULA FOR MEAN CURVATURE ONE SURFACES IN HYPERBOLIC THREE-SPACE AND IN DE SITTER THREE-SPACE

  • Yang, Seong-Deog
    • 대한수학회보
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    • 제54권1호
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    • pp.159-175
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    • 2017
  • We solve the $Bj{\ddot{o}}rling$ problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve ${\gamma}$ and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to ${\gamma}$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains ${\gamma}$ and the unit normal of which on ${\gamma}$ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.

On Weakly Z Symmetric Spacetimes

  • De, Uday Chand
    • Kyungpook Mathematical Journal
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    • 제58권4호
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    • pp.761-779
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    • 2018
  • The object of the present paper is to study weakly Z symmetric spacetimes $(WZS)_4$. At first we prove that a weakly Z symmetric spacetime is a quasi-Einstein spacetime and hence a perfect fluid spacetime. Next, we consider conformally flat $(WZS)_4$ spacetimes and prove that such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field ${\rho}$. We also study $(WZS)_4$ spacetimes with divergence free conformal curvature tensor. Moreover, we characterize dust fluid and viscous fluid $(WZS)_4$ spacetimes. Finally, we construct an example of a $(WZS)_4$ spacetime.

𝜂-RICCI SOLITONS ON 𝜖 - LP-SASAKIAN MANIFOLDS WITH A QUARTER-SYMMETRIC METRIC CONNECTION

  • Haseeb, Abdul;Prasad, Rajendra
    • 호남수학학술지
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    • 제41권3호
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    • pp.539-558
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    • 2019
  • In this paper, we study ${\eta}$-Ricci solitons on ${\epsilon}$-LP-Sasakian manifolds with a quarter-symmetric metric connection satisfying certain curvature conditions. In particular, we have discussed that the Ricci soliton on ${\epsilon}$-LP-Sasakian manifolds with a quarter-symmetric metric connection satisfying certain curvature conditions is expanding or steady according to the vector field ${\xi}$ being timelike or spacelike. Moreover, we construct 3-dimensional examples of an ${\epsilon}$-LP-Sasakian manifold with a quarter-symmetric metric connection to verify some results of the paper.