Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres

Karsten Grove; Luigi Verdiani; Burkhard Wilking; Wolfgang Ziller

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 2, page 159-170
  • ISSN: 0391-173X

Abstract

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In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case when both are equal to two, where existence of non-negatively curved metrics is known.

How to cite

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Grove, Karsten, et al. "Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.2 (2006): 159-170. <http://eudml.org/doc/239592>.

@article{Grove2006,
abstract = {In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case when both are equal to two, where existence of non-negatively curved metrics is known.},
author = {Grove, Karsten, Verdiani, Luigi, Wilking, Burkhard, Ziller, Wolfgang},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {159-170},
publisher = {Scuola Normale Superiore, Pisa},
title = {Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres},
url = {http://eudml.org/doc/239592},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Grove, Karsten
AU - Verdiani, Luigi
AU - Wilking, Burkhard
AU - Ziller, Wolfgang
TI - Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 2
SP - 159
EP - 170
AB - In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case when both are equal to two, where existence of non-negatively curved metrics is known.
LA - eng
UR - http://eudml.org/doc/239592
ER -

References

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  6. [GZ2] K. Grove and W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), 619–646. Zbl1038.53034MR1923478
  7. [HH] W. C. Hsiang and W. Y. Hsiang, On compact subgroups of the diffeomorphism groups of Kervaire spheres, Ann. of Math. 85 (1967), 359–369. Zbl0152.40604MR214083
  8. [Se] C. Searle, Cohomogeneity and positive curvature in low dimensions, Math. Z. 214 (1993), 491–498: Err. ibet. 226 (1997), 165–167. Zbl0804.53057MR1245208
  9. [ST] L. Schwachöfer and W. Tuschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), 59–91. Zbl1062.53027MR2091679
  10. [Wa] N. Wallach, Minimal immersions of symmetric spaces into spheres, In: “Symmetric Spaces”, Pure and Appl. Math., Vol. 8 , Marcel Dekker, New York, 1972, 1–40. Zbl0232.53027MR407774

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