# zbMATH — the first resource for mathematics

Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres. (English) Zbl 1170.53307
Summary: 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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
Full Text:
##### References:
 [1] A. V. Alekseevski and D. V. Alekseevski, $$G$$-manifolds with one dimensional orbit space, Adv. in Sov. Math. 8 (1992), 1-31. Zbl0769.57021 MR1155662 · Zbl 0769.57021 [2] A. Besse, “Einstein Manifolds”, Erg. der Math. und ihr. Grenz. Vol. 10, Springer Verlag, Berlin, 1987. Zbl0613.53001 MR867684 · Zbl 0613.53001 [3] A. Back and W. Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987), 207-227. Zbl0632.53048 MR906813 · Zbl 0632.53048 · doi:10.2307/2000711 [4] G. E. Bredon, “Introduction to Compact Transformation Groups”, Pure and Applied Mathematics (46), Academic Press, New York, 1972. Zbl0246.57017 MR413144 · Zbl 0246.57017 [5] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. 152 (2000), 331-367. Zbl0991.53016 MR1792298 · Zbl 0991.53016 · doi:10.2307/2661385 · www.math.princeton.edu · eudml:121963 [6] K. Grove and W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), 619-646. Zbl1038.53034 MR1923478 · Zbl 1038.53034 · doi:10.1007/s002220200225 [7] W. C. Hsiang and W. Y. Hsiang, On compact subgroups of the diffeomorphism groups of Kervaire spheres, Ann. of Math. 85 (1967), 359-369. Zbl0152.40604 MR214083 · Zbl 0152.40604 · doi:10.2307/1970349 [8] C. Searle, Cohomogeneity and positive curvature in low dimensions, Math. Z. 214 (1993), 491-498: Err. ibet. 226 (1997), 165-167. Zbl0804.53057 MR1245208 · Zbl 0804.53057 · doi:10.1007/BF02572419 · eudml:174583 [9] L. Schwachöfer and W. Tuschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), 59-91. Zbl1062.53027 MR2091679 · Zbl 1062.53027 · doi:10.1007/s00208-004-0538-x [10] 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. Zbl0232.53027 MR407774 · Zbl 0232.53027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.