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SCALAR CURVATURE, KILLING VECTOR FIELDS AND HARMONIC ONE-FORMS ON COMPACT RIEMANNIAN MANIFOLDS

Published online by Cambridge University Press:  24 August 2004

ZEJUN HU  and
HAIZHONG LI
ZEJUN HU
Affiliation:
Department of Mathematics, Zhengzhou University, 450052, Zhengzhou, Henan, People's Republic of China, huzj@zzu.edu.cn
HAIZHONG LI
Affiliation:
Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, People's Republic of Chinahli@math.tsinghua.edu.cn
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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.

Keywords

53C20 (primary)53C24 (secondary)
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 36 , Issue 5 , September 2004 , pp. 587 - 598
Copyright
© The London Mathematical Society 2004

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