The spectral geometry of the canonical Riemannian submersion of a compact Lie Group

Corey Dunn, Peter B. Gilkey, and JeongHyeong Park

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Submission date: 17. Oct. 2006
Pages: 13
published in: Journal of geometry and physics, 57 (2007) 10, p. 2065-2076 
DOI number (of the published article): 10.1016/j.geomphys.2007.05.002
Bibtex
MSC-Numbers: 58G25
Keywords and phrases: Riemannian submersion, Eigenform of the Laplacian, Finite Fourier Series
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Abstract:
Let G be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy be the multiplication operator. We show the associated fibration m from GxG to G is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull back of an eigenform on the base has finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non zero. We give necessary and sufficient conditions that the pull back of a form on the base is harmonic on the total space.