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(1999 / vol.2 / no.1)
Curvature, Spectra, and Riemannian Submersions
Peter B. Gilkey, Jeonghyeong Park
Pages. 164-169     



Let $Delta_M^p$ be the Laplacian on $p$ forms on a
closed Riemannian manifold $M$. Let $pi:Z
ightarrow Y$ be a
Riemannian submersion. We give necessary and sufficient conditions to
ensure the pull back of every eigen $p$ form over $Y$ is an eigen $p$ form
over $Z$; in this setting eigenvalues can not change so
$mu(lambda)=lambda$. We also study the holomorphic and the spinor
settings. We show that except in the spin context if a single eigen-section is
preserved, then the associated eigenvalue can not decrease. We also show
that in many contexts an eigenvalue can change.



1. Introduction
2. The real Laplacian
3. The complex Laplacian
4. The spin Laplacian



Dolbeault Laplacian, Riemannian Submersion, Eigenvalues, Spectra, Bochner Laplacian, Spin Laplacian



58G25