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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A Weitzenböck formula for canonical metrics on four-manifolds
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by Peng Wu PDF
Trans. Amer. Math. Soc. 369 (2017), 1079-1096 Request permission


We first provide an alternative proof of the classical Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds. As applications, we classify Einstein four-manifolds and conformally Einstein four-manifolds with half two-nonnegative curvature operator, which in some sense provides a characterization of Kähler-Einstein metrics and Hermitian, Einstein metrics with positive scalar curvature on four-manifolds, respectively. We also discuss the classification of four-dimensional gradient shrinking Ricci solitons with half two-nonnegative curvature operator and half harmonic Weyl curvature.
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Additional Information
  • Peng Wu
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
  • Address at time of publication: Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
  • MR Author ID: 845776
  • Email:
  • Received by editor(s): February 9, 2015
  • Published electronically: July 26, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 1079-1096
  • MSC (2010): Primary 53C25; Secondary 53C24
  • DOI:
  • MathSciNet review: 3572265