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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Eigenvalues of the form valued Laplacian
for Riemannian submersions


Authors: Peter B. Gilkey, John V. Leahy and Jeong Hyeong Park
Journal: Proc. Amer. Math. Soc. 126 (1998), 1845-1850
MSC (1991): Primary 58G25
DOI: https://doi.org/10.1090/S0002-9939-98-04733-9
MathSciNet review: 1485476
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\pi :Z\rightarrow Y$ be a Riemannian submersion of closed manifolds. Let $\Phi _{p}$ be an eigen $p$-form of the Laplacian on $Y$ with eigenvalue $\lambda $ which pulls back to an eigen $p$-form of the Laplacian on $Z$ 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 $Y$ is simply connected, then $\lambda =\mu $, so the eigenvalue does not change.


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Additional Information

Peter B. Gilkey
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: gilkey@math.uoregon.edu

John V. Leahy
Email: leahy@math.uoregon.edu

Jeong Hyeong Park
Affiliation: Department of Mathematics, Honam University, Seobongdong 59, Kwangsanku, Kwangju, 506-090 South Korea
Email: jhpark@honam.honam.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-98-04733-9
Keywords: Riemannian submersion, eigenvalues, Laplacian
Received by editor(s): May 20, 1996
Additional Notes: The first author’s research was partially supported by the NSF (USA); the third author’s, by BSRI-96-1425, the Korean Ministry of Education
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society