Journal of Geometry

, Volume 98, Issue 1–2, pp 91–113 | Cite as

Critical metrics of the Schouten functional

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Abstract

Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L 2-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics.

Mathematics Subject Classification (2010)

58E11 53C20 53C25 53C55 

Keywords

Schouten tensor Schouten functional conformally flat metric self-dual metric 
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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsZhengzhou UniversityZhengzhouPeople’s Republic of China
  2. 2.Mathematical InstituteTohoku UniversitySendaiJapan
  3. 3.Institut für Mathematik, MA 8-3Technische Universität BerlinBerlinGermany

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