Abstract
We classify those manifolds of positive Euler characteristic on which a Lie group G acts with cohomogeneity one, where G is classical simple.
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A. V. Alekseevskiǐ, D. V. Alekseevskiǐ, G-manifolds with one dimensional orbit space, in: Lie Groups, their Discrete Subgroups, and Invariant Theory, Advances in Soviet Mathematics, Amer. Math. Soc., Providence RI, 1992, pp. 1–31.
A. V. Alekseevsky, D. V. Alekseevsky, Riemannian G-manifolds with one dimensional orbit space, Ann. Global Anal. Geom. 11 (1993), no. 3, 197–211.
D. V. Alekseevsky, Riemannian manifolds of cohomogeneity one, in: Differential Geometry and its Applications, Eger 1989, Colloq. Math. Soc. János Bolyai, Vol. 56, North-Holland, Amsterdam, 1992, pp. 9–22.
D. V. Alekseevsky, F. Podestá, Compact cohomogeneity one Riemannian manifolds of positive Euler characteristic and quaternionic Kähler manifolds, in: Geometry, Topology, and Physics, Campinas, 1996, de Gruyter, Berlin, 1997, pp. 1–33.
G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972. Russian transl.: Г. Бредон, Введенuе в mеорuю комnакmных груnn nреобразанцй, Наука, М., 1980.
M. Goto, F. D. Grosshans, Semisimple Lie Algebras, Marcel Dekker, New York, 1978. Russian transl.: Гото, Ф. Гроссханс, Полуnросmые алгебры Лц, Мир, М., 1981.
K. Grove, B. Wilking, W. Ziller, Positively curved cohomogeneity one manifolds and 3-Sasakian manifolds, J. Diff. Geom. 78 (2008), no. 1. 33–111.
K. Grove, W. Ziller, Curvature and symmetry of Milnor spheres, Ann. Math. 152 (2000), 331–367.
K. Grove, W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Inv. Math. 149 (2002), 619–646.
K. Grove, L. Verdiani, B. Wilking, W. Ziller, Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres, Ann. Sc. Norm. Super. Pisa Cl. Sci. 5 (2006), 159–170.
C. Hoelscher, Classification of cohomogeneity one manifolds in low dimensions, Pacific J. Math. 125 (2010), no. 1, 129–185.
P. S. Mostert, On a compact Lie group acting on a manifold, Ann. Math. 65 (1957), 447–455.
W. D. Neumann, 3-dimensional G-manifolds with 2-dimensional orbits, in Proc. Conf. on Transformation Groups, 1968, 220–222.
J. Parker, 4-dimensional G-manifolds with 3-dimensional orbits, Pacific J. Math. 125 (1986), no. 1, 187–204.
H.-C. Wang, Homogeneous spaces with non-vanishing Euler characteristics, Ann. Math. 50 (1949), 925–953.
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Frank, P. Cohomogeneity one manifolds with positive Euler characteristic. Transformation Groups 18, 639–684 (2013). https://doi.org/10.1007/s00031-013-9227-8
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DOI: https://doi.org/10.1007/s00031-013-9227-8