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Transactions of the American Mathematical Society

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

 
 

 

A warped product version of the Cheeger-Gromoll splitting theorem


Author: William Wylie
Journal: Trans. Amer. Math. Soc. 369 (2017), 6661-6681
MSC (2010): Primary 53C20
DOI: https://doi.org/10.1090/tran/7003
Published electronically: May 16, 2017
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Abstract: We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $ CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $ CD(0,1)$, we show that the fundamental group of $ M$ is the fundamental group of a compact manifold with non-negative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is $ CD(0,1)$.


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Additional Information

William Wylie
Affiliation: Department of Mathematics, 215 Carnegie Building, Syracuse University, Syracuse, New York 13244
Email: wwylie@syr.edu

DOI: https://doi.org/10.1090/tran/7003
Received by editor(s): June 23, 2015
Received by editor(s) in revised form: January 7, 2016, April 4, 2016, and June 16, 2016
Published electronically: May 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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