Curvature and symmetry of Milnor spheres.

*(English)*Zbl 0991.53016It is still not well understood which compact manifolds carry metrics of positive sectional curvature. On the one hand only very few obstructions against such metrics are known which are not at the same time obstructions against much weaker conditions such as positive Ricci curvature or positive scalar curvature. On the other hand, not many examples of compact Riemannian manifolds of positive sectional curvature are known either. For example, it is still open whether or not any of the exotic spheres has a metric of positive sectional curvature.

Since these questions seem at present out of reach the paper addresses to existence of nonnegatively curved metrics. One of the main results says that 10 of the 14 exotic 7-spheres carry metrics of nonnegative sectional curvature. Before the present work only one such exotic sphere was known, the Gromoll-Meyer sphere [D. Gromoll and W. Meyer, Ann. Math. (2) 100, 401-406 (1974; Zbl 0293.53015)]. These 10 spheres are exactly those which can be exhibited as \(S^3\)-bundles over \(S^4\), the so-called Milnor spheres.

The authors show more generally that the total space of every vector bundle and every sphere bundle over \(S^4\) admits a complete Riemannian metric of nonnegative sectional curvature. The metrics constructed here are metrics of cohomogeneity one, i.e. their isometry groups have orbits of codimension one.

One of the key results of the paper says that every manifold with a cohomogeneity one action whose singular orbits have codimension two carries an invariant metric of nonnegative sectional curvature. It is an open conjecture that this still holds with the condition on the codimension of the singular orbits removed.

As another application of their method the authors show that each of the 4 oriented diffeomorphism types of 5-manifolds homotopy equivalent to \(\mathbb{R}\mathbb{P}^5\) admits metrics with \(K\geq 0\). Moreover, the authors construct infinitely many almost free SO(3)-actions on \(S^7\) (preserving the Hopf fibration structure) and on all of the exotic Milnor 7-spheres.

Since these questions seem at present out of reach the paper addresses to existence of nonnegatively curved metrics. One of the main results says that 10 of the 14 exotic 7-spheres carry metrics of nonnegative sectional curvature. Before the present work only one such exotic sphere was known, the Gromoll-Meyer sphere [D. Gromoll and W. Meyer, Ann. Math. (2) 100, 401-406 (1974; Zbl 0293.53015)]. These 10 spheres are exactly those which can be exhibited as \(S^3\)-bundles over \(S^4\), the so-called Milnor spheres.

The authors show more generally that the total space of every vector bundle and every sphere bundle over \(S^4\) admits a complete Riemannian metric of nonnegative sectional curvature. The metrics constructed here are metrics of cohomogeneity one, i.e. their isometry groups have orbits of codimension one.

One of the key results of the paper says that every manifold with a cohomogeneity one action whose singular orbits have codimension two carries an invariant metric of nonnegative sectional curvature. It is an open conjecture that this still holds with the condition on the codimension of the singular orbits removed.

As another application of their method the authors show that each of the 4 oriented diffeomorphism types of 5-manifolds homotopy equivalent to \(\mathbb{R}\mathbb{P}^5\) admits metrics with \(K\geq 0\). Moreover, the authors construct infinitely many almost free SO(3)-actions on \(S^7\) (preserving the Hopf fibration structure) and on all of the exotic Milnor 7-spheres.

Reviewer: Christian Bär (Hamburg)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |