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On the principal Ricci curvatures of a Riemannian 3-manifold

  • Amir Babak Aazami EMAIL logo and Charles M. Melby-Thompson
From the journal Advances in Geometry


We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.

MSC 2010: 53C20; 53C22; 53C25
  1. Communicated by: P. Eberlein

  2. Funding: This work was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan; this manuscript first appeared when both authors were members of the Kavli Institute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo, Japan.


We kindly thank Benjamin Schmidt, Wolfgang Ziller, and Robert Bryant for very helpful discussions.


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Received: 2017-05-26
Published Online: 2018-07-20
Published in Print: 2019-04-24

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