New examples of obstructions to non-negative sectional curvatures in cohomogeneity one manifolds

He, Chenxu

doi:10.1090/S0002-9947-2014-06194-1

New examples of obstructions to non-negative sectional curvatures in cohomogeneity one manifolds
HTML articles powered by AMS MathViewer

by Chenxu He PDF

Trans. Amer. Math. Soc. 366 (2014), 6093-6118 Request permission

Abstract:

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classify all class one representations for a pair $(G,H)$ with $G/H$ a sphere, which are used to construct the examples.

References

A. V. Alekseevskiĭ and D. V. Alekseevskiĭ, $G$-manifolds with one-dimensional orbit space, Lie groups, their discrete subgroups, and invariant theory, Adv. Soviet Math., vol. 8, Amer. Math. Soc., Providence, RI, 1992, pp. 1–31. MR 1155662, DOI 10.1007/bf01084048
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
Jeff Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geometry 8 (1973), 623–628. MR 341334
Karsten Grove, Burkhard Wilking, and Wolfgang Ziller, Positively curved cohomogeneity one manifolds and 3-Sasakian geometry, J. Differential Geom. 78 (2008), no. 1, 33–111. MR 2406265
Karsten Grove, Luigi Verdiani, Burkhard Wilking, and Wolfgang Ziller, Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 159–170. MR 2244696
Karsten Grove and Wolfgang Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), no. 1, 331–367. MR 1792298, DOI 10.2307/2661385
Karsten Grove and Wolfgang Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. Math. 149 (2002), no. 3, 619–646. MR 1923478, DOI 10.1007/s002220200225
Chenxu He, Non-negatively curved cohomogeneity one manifolds, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of Pennsylvania. MR 2713860
C. He, New examples of obstructions to non-negative sectional curvatures in cohomogeneity one manifolds, arXiv math:DG/0910. 5712v1, 2009.
Eldar Straume, Compact connected Lie transformation groups on spheres with low cohomogeneity. I, Mem. Amer. Math. Soc. 119 (1996), no. 569, vi+93. MR 1297539, DOI 10.1090/memo/0569
N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 1, Mathematics and its Applications (Soviet Series), vol. 72, Kluwer Academic Publishers Group, Dordrecht, 1991. Simplest Lie groups, special functions and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1143783, DOI 10.1007/978-94-011-3538-2
Nolan R. Wallach, Minimal immersions of symmetric spaces into spheres, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 1–40. MR 0407774
Burkhard Wilking, A duality theorem for Riemannian foliations in nonnegative sectional curvature, Geom. Funct. Anal. 17 (2007), no. 4, 1297–1320. MR 2373019, DOI 10.1007/s00039-007-0620-0
Wolfgang Ziller, Examples of Riemannian manifolds with non-negative sectional curvature, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 63–102. MR 2408264, DOI 10.4310/SDG.2006.v11.n1.a4