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Nonnegatively curved fixed point homogeneous 5-manifolds. (English) Zbl 1239.53046
Ann. Global Anal. Geom. 41, No. 2, 253-263 (2012); erratum ibid. 45, No. 2, 151-153 (2014).
Let \((M,g)\) be a closed Riemannian manifold endowed with an effective smooth action by a compact Lie group \(G\). If the action has fixed points, then \(\dim(M/G)\) is bounded below by the dimension of the fixed point set and one defines the fixed point cohomogeneity to be \[ \text{cohomfix}(M,G):=\dim(M/G)-\dim(\text{Fix}(M,g))-1\geq0\,. \] If the fixed point cohomogeneity of the action is \(0\), the action is said to be fixed point homogeneous and \((M,g)\) is said to be a fixed point homogeneous manifold; in this setting, the fixed point set has codimension 1 in the orbit space. The authors show: \medbreakMain Theorem: Let \(M^5\) be a closed, simply connected, \(5\)-dimensional nonnegatively curved fixed point homogeneous \(G\)-manifold. Then \(G\) is one of the groups \(\{SO(5), SO(4), SU(2), SO(3), S^1\}\) and one has the following classification: \smallbreak(a) If \(G\in\{SO(5),SO(4),SU(2)\}\), then \(M\) is diffeomorphic to \(S^5\). \smallbreak(b) If \(G\in\{SO(3),S^1\}\), then \(M\) is diffeomorphic to \(S^5\) or to one of the two bundles over \(S^2\) with fiber \(S^3\). \medbreak The authors note that the list of fixed point homogeneous 5-manifolds in the main theorem contains every known closed simply connected \(5\)-manifold of nonnegative sectional curvature except for the Wu manifold \(SU(3)/SO(3)\). Section 1 of the paper contains an introduction to the subject at hand. In Section 2, basic facts about group actions and Alexandrov spaces are recalled. Section 3 contains the proof of the main theorem; the cases \(\{SO(5),SO(4),SU(2),SO(3)\}\) are treated using standard classification results. The case \(G=S^1\) has to be treated separately; the hypothesis of nonnegative curvature enables the authors to show by looking at the orbit space structure that \(M^5\) decomposes as the union of two disk bundles over smooth submanifolds of \(M^5\) one of which is a 3-dimensional component of the fixed point set; after examining \(H_2(M^5;\mathbb Z)\), the conclusion follows from the Barden-Smale classification of smooth closed simply connected 5-manifolds [D. Barden, Ann. Math. (2) 82, 365–385 (1965; Zbl 0136.20602); S. Smale, Ann. Math. (2) 75, 38–46 (1962; Zbl 0101.16103)].

MSC:
53C20 Global Riemannian geometry, including pinching
57S15 Compact Lie groups of differentiable transformations
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