Explicit form of Laplace-Beltrami operator on SO(3) in the view of Fourier analysis

Research Article
Open access

Explicit form of Laplace-Beltrami operator on SO(3) in the view of Fourier analysis

Yigao Feng 1*
  • 1 Beijing Institute of Technology, Zhuhai    
  • *corresponding author 13286055168@bitzh.edu.cn
Published on 17 November 2023 | https://doi.org/10.54254/2753-8818/10/20230325
TNS Vol.10
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-131-5
ISBN (Online): 978-1-83558-132-2

Abstract

Fourier analysis plays a central role in the modern physics, engineering, and mathematics itself. In the field of differential geometry, a Lie group G gives a symmetric structure, and one may apply the Fourier analysis by means of matrix-valued irreducible representations. Even though the entries of these irreducible representations are already shown to be the eigenfunctions of the Laplace-Beltrami operator, it is still desirable to consider a concrete example where both the operator and the irreducible representations can be computed explicitly. This study gives an explicit form of the Laplace-Beltrami operator on SO(3) using direct computations and show also that each entry of the irreducible representations o_n^ij is indeed an eigenfunction of this operator. Therefore, one can also find the application of the Fourier Analysis on differential equations, in this study Poisson’s equation as an example, using the Laplace-Beltrami operator as the corresponding differential operator. Overall, these results shed light on guiding further exploration of Fourier analysis.

Keywords:

Fourier analysis on SO(3), Laplace-Beltrami operator on SO(3), poisson’s equation

Feng,Y. (2023). Explicit form of Laplace-Beltrami operator on SO(3) in the view of Fourier analysis. Theoretical and Natural Science,10,107-114.
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References

[1]. Elias M S and Rami S 2007 Fourier Analysis Princeton University Press.

[2]. Alexander G 2020 Analysis on manifolds and volume growth Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs (Advances in Analysis and Geometry vol 3) ed. Alexander G and Yuhua S pp 299-324.

[3]. Alexander G, Yuri N and Yau S T 2004 Eigenvalues of elliptic operators and geometric applications Eigenvalues of Laplacians and Other Geometric Operators (Surveys in Differential Geometry vol 9) ed Alexander G and Yau S T pp 147-218.

[4]. Sugiura M 1971 Fourier series of smooth functions on compact Lie groups Osaka J. Math pp 33-47.

[5]. Dym H and McKean H P 1972 Fourier Series and Integrals. Academic Press.

[6]. Vilenkin N J 1968 Special Functions and the Theory of Group Representations. English translation: American mathematical society.

[7]. Elias M S and Rami S 2005. Real Analysis. Princeton University Press.

[8]. John M L 2018 Introduction to Riemannian Manifolds Springer.

[9]. Jean G and Jocelyn Q 2020 Differential Geometry and Lie Groups: A Computational Perspective Springer p 417

[10]. Duistermaat J J and Kolk J 2020 Lie Groups Springer.

Cite this article

Feng,Y. (2023). Explicit form of Laplace-Beltrami operator on SO(3) in the view of Fourier analysis. Theoretical and Natural Science,10,107-114.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN：978-1-83558-131-5(Print) / 978-1-83558-132-2(Online)
Editor：Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.10
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Elias M S and Rami S 2007 Fourier Analysis Princeton University Press.

[2]. Alexander G 2020 Analysis on manifolds and volume growth Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs (Advances in Analysis and Geometry vol 3) ed. Alexander G and Yuhua S pp 299-324.

[3]. Alexander G, Yuri N and Yau S T 2004 Eigenvalues of elliptic operators and geometric applications Eigenvalues of Laplacians and Other Geometric Operators (Surveys in Differential Geometry vol 9) ed Alexander G and Yau S T pp 147-218.

[4]. Sugiura M 1971 Fourier series of smooth functions on compact Lie groups Osaka J. Math pp 33-47.

[5]. Dym H and McKean H P 1972 Fourier Series and Integrals. Academic Press.

[6]. Vilenkin N J 1968 Special Functions and the Theory of Group Representations. English translation: American mathematical society.

[7]. Elias M S and Rami S 2005. Real Analysis. Princeton University Press.

[8]. John M L 2018 Introduction to Riemannian Manifolds Springer.

[9]. Jean G and Jocelyn Q 2020 Differential Geometry and Lie Groups: A Computational Perspective Springer p 417

[10]. Duistermaat J J and Kolk J 2020 Lie Groups Springer.

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