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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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