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$$S^1$$-actions on $$4$$-manifolds and fixed point homogeneous manifolds of nonnegative curvature. (English) Zbl 1297.53006
Münster: Univ. Münster, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik und Informatik (Diss.). ix, 78 p. (2014).
Publisher’s description: Let $$M$$ be a closed, nonnegatively curved and simply connected Riemannian 4-manifold equipped with an isometric action of the circle $$\mathsf{S}^1$$ with only isolated fixed points. The first main theorem of this thesis shows that there exists a sequence of smooth, positively curved metrics on the quotient space $$M/\mathsf{S}^1$$ that converges in Gromov-Hausdorff topology to the singular quotient metric of $$M/\mathsf{S}^1$$, and the same holds for the two fold branched cover of $$M/\mathsf{S}^1$$ along a singular closed curve. This result leads to a solely geometric proof (not appealing to the Poincaré conjecture) of the equivariant classification theorem for nonnegatively curved, simply connected 4-manifolds with circle symmetry obtained by Grove and Wilking in their preprint “A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry”.
The second main theorem deals with fixed point homogeneous manifolds. These are by definition Riemannian manifolds admitting an effective, isometric action by a Lie group $$\mathsf{G}$$ with nonempty fixed point set such that a fixed point component has codimension 1 in the orbit space $$M/\mathsf{G}$$. It is shown that a closed, nonnegatively curved, fixed point homogeneous $$\mathsf{G}$$-manifold admits a double disk bundle decomposition over a fixed point component of maximal dimension and another smooth $$\mathsf{G}$$-invariant submanifold of $$M$$. As an application of this result it is shown that a closed, simply connected, nonnegatively curved torus manifold is rationally elliptic.

MSC:
 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C30 Differential geometry of homogeneous manifolds
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