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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 15 (1961), 179–246 (French). MR 133083 ---, Trois remarques sur les variétés riemannienes à courbure positive, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A76-A78. MR 33 #7966.
- Nolan R. Wallach, Homogeneous positively pinched Riemannian manifolds, Bull. Amer. Math. Soc. 76 (1970), 783–786. MR 257935, DOI https://doi.org/10.1090/S0002-9904-1970-12551-4
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53.72
Retrieve articles in all journals with MSC: 53.72
Additional Information
Keywords:
Invariant Riemannian metric,
family of Riemannian metrics,
positive sectional curvature,
homogeneous space,
Haar measure
Article copyright:
© Copyright 1970
American Mathematical Society