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Eigenvalues of the form valued Laplacian for Riemannian submersions

Authors:	Peter B Gilkey John V Leahy Jeong Hyeong Park

Institution:	Department of Mathematics, University of Oregon, Eugene, Oregon 97403 ; Department of Mathematics, University of Oregon, Eugene, Oregon 97403 ; Department of Mathematics, Honam University, Seobongdong 59, Kwangsanku, Kwangju, 506-090 South Korea

Abstract:	Let $\pi :Z\rightarrow Y$ be a Riemannian submersion of closed manifolds. Let $\Phi _{p}$ be an eigen -form of the Laplacian on with eigenvalue $\lambda$ which pulls back to an eigen -form of the Laplacian on with eigenvalue $\mu$ . We are interested in when the eigenvalue can change. We show that $\lambda \le \mu$ , so the eigenvalue can only increase; and we give some examples where $\lambda <\mu$ , so the eigenvalue changes. If the horizontal distribution is integrable and if is simply connected, then $\lambda =\mu$ , so the eigenvalue does not change.

Keywords:	Riemannian submersion eigenvalues Laplacian

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