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.

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.

Classification : 53C20 
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     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},
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     url = {http://archive.numdam.org/item/ASNSP_2006_5_5_2_159_0/}
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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://archive.numdam.org/item/ASNSP_2006_5_5_2_159_0/

[AA] A. V. Alekseevski and D. V. Alekseevski, G-manifolds with one dimensional orbit space, Adv. in Sov. Math. 8 (1992), 1-31. | MR | Zbl

[Be] A. Besse, “Einstein Manifolds”, Erg. der Math. und ihr. Grenz. Vol. 10, Springer Verlag, Berlin, 1987. | MR | Zbl

[BH] A. Back and W. Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987), 207-227. | MR | Zbl

[Br] G. E. Bredon, “Introduction to Compact Transformation Groups”, Pure and Applied Mathematics (46), Academic Press, New York, 1972. | MR | Zbl

[GZ1] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. 152 (2000), 331-367. | MR | Zbl

[GZ2] K. Grove and W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), 619-646. | MR | Zbl

[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. | MR | Zbl

[Se] C. Searle, Cohomogeneity and positive curvature in low dimensions, Math. Z. 214 (1993), 491-498: Err. ibet. 226 (1997), 165-167. | MR | Zbl

[ST] L. Schwachöfer and W. Tuschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), 59-91. | MR | Zbl

[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. | MR | Zbl