A comparison theorem for eigenvalues of the covariant Laplacian. (English) Zbl 0716.58030
The minimax principle is employed to bound the eigenvalues \(\lambda_ n\), of a covariant Laplacian associated to a connection on a vector bundle, above by the eigenvalues \(\mu_ n\), of a corresponding scalar Laplacian. If there exists a non-vanishing eigensection, of the bundle, and \(n=0\), then equality is attained. Some examples are given.
Reviewer: H.Donnelly
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 53C20 | Global Riemannian geometry, including pinching |
Keywords:
Riemannian manifold; minimax principle; eigenvalues; covariant Laplacian; connection; vector bundleReferences:
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