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

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 Poissons 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.