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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Wissenschaftlicher Beirat: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
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Mathematical Citation Quotient (MCQ) 2018: 0.53

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1615-7168
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Band 19, Heft 2

Hefte

On the principal Ricci curvatures of a Riemannian 3-manifold

Amir Babak Aazami / Charles M. Melby-Thompson
Online erschienen: 20.07.2018 | DOI: https://doi.org/10.1515/advgeom-2018-0020

Abstract

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.

Keywords: Riemannian 3-manifold; Ricci curvature

MSC 2010: 53C20; 53C22; 53C25

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Artikelinformationen

Erhalten: 26.05.2017

Online erschienen: 20.07.2018

Erschienen im Druck: 24.04.2019


Communicated by: P. Eberlein

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.


Quellenangabe: Advances in Geometry, Band 19, Heft 2, Seiten 251–262, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0020.

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