an:06006617
Zbl 1239.53046
Galaz-Garcia, Fernando; Spindeler, Wolfgang
Nonnegatively curved fixed point homogeneous 5-manifolds
EN
Ann. Global Anal. Geom. 41, No. 2, 253-263 (2012); erratum ibid. 45, No. 2, 151-153 (2014).
00295114
2012
j
53C20 57S15
nonnegative curvature; circle action; 5 manifold; fixed point homogeneous; Alexandrov spaces; Barden-Smale classification; Soul theorem; double soul theorem
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 \textit{fixed point homogeneous} and \((M,g)\) is said to be a \textit{fixed point homogeneous manifold}; in this setting, the fixed point set has codimension 1 in the orbit space. The authors show: \medbreak\noindent\textbf{Main 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\noindent (a) If \(G\in\{SO(5),SO(4),SU(2)\}\), then \(M\) is diffeomorphic to \(S^5\). \smallbreak\noindent (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 [\textit{D. Barden}, Ann. Math. (2) 82, 365--385 (1965; Zbl 0136.20602); \textit{S. Smale}, Ann. Math. (2) 75, 38--46 (1962; Zbl 0101.16103)].
Peter B. Gilkey (Eugene)
Zbl 0136.20602; Zbl 0101.16103