Construction of new Lie group and its geometric properties

Muhammad Asad Iqbal; Abid Ali; Ibtesam Alshammari; Cenap Ozel; Muhammad Asad Iqbal; Abid Ali; Ibtesam Alshammari; Cenap Ozel

doi:10.3934/math.2024298

AIMS Mathematics

2024, Volume 9, Issue 3: 6088-6108. doi: 10.3934/math.2024298

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Research article

Construction of new Lie group and its geometric properties

1.
Department of Mathematics, Comsats University Islamabad, Pakistan; asadiqbal494@gmail.com
2.
Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
3.
University of Hafr Al Batin, Department of Mathematics, Hafr Al Batin, KSA; iealshamri@uhb.edu.sa
4.
University of King Abdulaziz, Department of Mathematics, Jeddah 21589, KSA; cozel@kau.edu.sa

Received: 12 December 2023 Revised: 18 January 2024 Accepted: 24 January 2024 Published: 02 February 2024
MSC : 22Exx, 57Sxx, 58-XX

In this paper, we constructed a novel Lie group by using oblate spheroidal coordinates. First, we took the metric tensor of oblate spheroidal coordinates, then found its Killing vectors by using the Killing equation. After solving a system of partial differential equations, we obtained the Killing vectors. With the help of these Killing vectors, we first constructed finite Lie algebra and then proved that Killing vectors form a Lie group. Also, we described the geometric properties in which this Lie group forms a regular surface, defined the differential map and differential of normal vector field, and found the gaussian and mean curvatures.
- Lie group $ G $,
- Lie algebra,
- Lie symmetry,
- Killing vectors $ X^1 $, $ X^2 $ and $ X^3 $
Citation: Muhammad Asad Iqbal, Abid Ali, Ibtesam Alshammari, Cenap Ozel. Construction of new Lie group and its geometric properties[J]. AIMS Mathematics, 2024, 9(3): 6088-6108. doi: 10.3934/math.2024298

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Abstract

In this paper, we constructed a novel Lie group by using oblate spheroidal coordinates. First, we took the metric tensor of oblate spheroidal coordinates, then found its Killing vectors by using the Killing equation. After solving a system of partial differential equations, we obtained the Killing vectors. With the help of these Killing vectors, we first constructed finite Lie algebra and then proved that Killing vectors form a Lie group. Also, we described the geometric properties in which this Lie group forms a regular surface, defined the differential map and differential of normal vector field, and found the gaussian and mean curvatures.

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