\(S^1\)-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.

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 |