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Ricci flow and curvature on the variety of flags on the two dimensional projective space over the complexes, quaternions and octonions


Authors: Man-Wai Cheung and Nolan R. Wallach
Journal: Proc. Amer. Math. Soc. 143 (2015), 369-378
MSC (2010): Primary 53C21; Secondary 53C20, 53C30
DOI: https://doi.org/10.1090/S0002-9939-2014-12241-6
Published electronically: August 18, 2014
MathSciNet review: 3272761
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Abstract: For homogeneous metrics on the spaces over the complexes, quaternions and octonions, it is shown that the Ricci flow can move a metric of strictly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature. A variant of the method of Böhm and Wilking is given, proving that one can flow a metric of positive sectional curvature to one with Ricci curvature of indefinite signature in the quaternionic and octonian cases. A proof is given that this cannot occur in the complex case.


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Additional Information

Man-Wai Cheung
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: m1cheung@ucsd.edu

Nolan R. Wallach
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: nwallach@ucsd.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12241-6
Keywords: Ricci curvature, Riemannian curvature, homogeneous space, Ricci flow
Received by editor(s): February 17, 2013
Published electronically: August 18, 2014
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society

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