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Low dimensional polar actions. (English) Zbl 1318.58008
A polar manifold \(M\) is a complete Riemannian \(G\)-manifold with a complete submanifold \(\Sigma\) that orthogonally intersects every \(G\)-orbit defined by the action of a compact connected Lie group \(G\). Such submanifold is named section of the polar manifold \(M\) and \(\Pi=N(\Sigma)/Z(\Sigma)\) the polar group, where \(N(\Sigma)=\{g\in G: g\Sigma=\Sigma\}\) and \(Z(\Sigma)=\{g\in G: gx=x,\forall x\in \Sigma\}\). \(M\) is assumed closed and simply-connected. The orbit space \(M/G\) is compact isometric to \(\Sigma/\Pi\) and this provides the orbit space an orbifold cover structure, an optimal dimension reduction and its stratification by isotropy types.
The author classifies all closed simply-connected polar manifolds of dimension up to \(5\), for almost-effective actions. The classification is by equivariant type of the \(G\) action and up to diffeomorphism of \(M\).
The most complex classification is the case of dimension 5, where the manifolds \(\mathbb{S}^3\times_k \mathbb{S}^2 =\mathbb{S}^3\times \mathbb{S}^3/S^1_{(1,-1,k,0)}\), with \(k\in \mathbb{Z}\), diffeomorphic to either \(\mathbb{S}^3\times \mathbb{S}^2\) or \(\mathbb{S}^3 \tilde{\times} \mathbb{S}^2\), are described as main examples, with actions induced from actions on \(\mathbb{S}^3\times \mathbb{S}^3\), as for example circle actions, \(\mathrm{SU}(2)\) or \(\mathrm{SU}(2)\times S^1\), \(T^3\) actions, all them with non-negative curvature. Effective polar \(T^2\) actions are described in Theorem A. Polar actions by an abelian Lie group are determined in Theorem B. In Theorem C almost-effective actions on \(M\) of cohomogeneity at least 2 by a compact connected non-abelian Lie group \(G\) are classified and in Theorem D the particular case \(M\) with invariant metric of non-negative sectional curvature. Polar \(S^1\)-actions on \(M\) of any dimension are explained in Proposition 3.2. For lower dimensions, Corollary 3.3 gives a uniqueness result for effective cohomogeneity 2 polar 3-manifolds, and Theorem 3.4 classifies all the 4-dimensional cases of cohomogeneity \(k\geq 2\).

MSC:
58D19 Group actions and symmetry properties
57M60 Group actions on manifolds and cell complexes in low dimensions
53C10 \(G\)-structures
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