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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Positively curved deformations of invariant Riemannian metrics


Author: Alan Weinstein
Journal: Proc. Amer. Math. Soc. 26 (1970), 151-152
MSC: Primary 53.72
DOI: https://doi.org/10.1090/S0002-9939-1970-0262977-8
MathSciNet review: 0262977
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Abstract: Let $ {K_\gamma }$ denote the sectional curvature function of the Riemannian metric $ \gamma $ on a manifold $ M$. Suppose $ M$ admits no metric $ \gamma $ invariant under the action of a compact group $ G$ and having $ {K_\gamma } > 0$. It is shown that a $ G$-invariant metric $ \gamma (0)$ with $ {K_{\gamma (0)}} \geqq 0$ cannot be embedded in a $ 1$-parameter family $ \gamma (t)$ for which $ {[d{K_{\gamma (t)}}/dt]_{t = 0}}$ is positive wherever $ {K_{\gamma (0)}}$ is zero.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0262977-8
Keywords: Invariant Riemannian metric, family of Riemannian metrics, positive sectional curvature, homogeneous space, Haar measure
Article copyright: © Copyright 1970 American Mathematical Society

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