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Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds

Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds Abstract It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal. 2000 Mathematics Subject Classification 53C20 (primary), 53C24 (secondary). © London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds

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References (8)

Publisher
Oxford University Press
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609304003455
Publisher site
See Article on Publisher Site

Abstract

Abstract It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal. 2000 Mathematics Subject Classification 53C20 (primary), 53C24 (secondary). © London Mathematical Society

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Sep 1, 2004

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