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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.
@article{ASNSP_2006_5_5_2_159_0,
author = {Grove, Karsten and Verdiani, Luigi and Wilking, Burkhard and Ziller, Wolfgang},
title = {Non-negative curvature obstructions in cohomogeneity one and the {Kervaire} spheres},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {159--170},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 5},
number = {2},
year = {2006},
mrnumber = {2244696},
zbl = {1170.53307},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_2_159_0/}
}
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 SP - 159 EP - 170 VL - 5 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_2_159_0/ LA - en ID - ASNSP_2006_5_5_2_159_0 ER -
%0 Journal Article %A Grove, Karsten %A Verdiani, Luigi %A Wilking, Burkhard %A Ziller, Wolfgang %T Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 159-170 %V 5 %N 2 %I Scuola Normale Superiore, Pisa %U http://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_2_159_0/ %G en %F ASNSP_2006_5_5_2_159_0
Grove, Karsten; Verdiani, Luigi; Wilking, Burkhard; Ziller, Wolfgang. Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 159-170. http://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_2_159_0/