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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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  • [1] M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 179-246. MR 24 #A2919. MR 0133083 (24:A2919)
  • [2] -, Trois remarques sur les variétés riemannienes à courbure positive, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A76-A78. MR 33 #7966.
  • [3] N. Wallach, Homogeneous positively pinched riemannian manifolds, (to appear). MR 0257935 (41:2584)

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Additional Information

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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