On differential analysis of spacelike flows on normal congruence of surfaces

Melek Erdoğdu; Ayșe Yavuz; Melek Erdoğdu; Ayșe Yavuz

doi:10.3934/math.2022753

AIMS Mathematics

2022, Volume 7, Issue 8: 13664-13680. doi: 10.3934/math.2022753

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Research article

On differential analysis of spacelike flows on normal congruence of surfaces

Melek Erdoğdu ^{1
,
,},
Ayșe Yavuz ²

1.
Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya 42090, Turkey
2.
Department of Mathematics and Science Education, Necmettin Erbakan University, Konya 42090, Turkey

Received: 18 January 2022 Revised: 13 May 2022 Accepted: 16 May 2022 Published: 23 May 2022
MSC : 53A35, 53A04, 53Z05

The present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the $ \operatorname{curl} $ of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both $ s-lines $ and $ b-lines. $ Also, we examine a normal congruence of surfaces containing the $ s-lines $ and $ b-lines $. By compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this normal congruence of surface. Intrinsic geometric properties of this normal congruence of surfaces are given.
- anholonomic coordinates,
- normal congruence,
- flows of spacelike curve,
- timelike surface,
- Gauss-Mainardi-Codazzi equations
Citation: Melek Erdoğdu, Ayșe Yavuz. On differential analysis of spacelike flows on normal congruence of surfaces[J]. AIMS Mathematics, 2022, 7(8): 13664-13680. doi: 10.3934/math.2022753

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Abstract

The present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the $ \operatorname{curl} $ of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both $ s-lines $ and $ b-lines. $ Also, we examine a normal congruence of surfaces containing the $ s-lines $ and $ b-lines $. By compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this normal congruence of surface. Intrinsic geometric properties of this normal congruence of surfaces are given.

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