Title: Curvature and Symmetry of Milnor Spheres
Authors: Karsten Grove and Wolfgang Ziller
E-Mail: kng@math.umd.edu , wziller@math.upenn.edu
In this paper we explore the geometry and topology of
cohomogeneity one manifolds, i.e. manifolds with a group action whose principal
orbits are hypersurfaces. We show that the principal group action of every
principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one
action.
As a consequence we prove that every vector bundle and every sphere bundle over
S^4 has a complete metric with non-negative curvature.
It is well known that 15 of the 27 exotic spheres in dimension 7 can be
written as S^3 bundles over
S^4 in infinitely many ways, and hence we obtain infinitely many non-negatively
curved metrics on these exotic spheres.
A further consequence will be that there are infinitely many almost free actions
by SO(3) on S^7, i.e. all isotropy groups are finite.
These actions preserve the Hopf fibration S^3 -> S^7 -> S^4 but do
not extend to the disc D^8. We also construct infinitely many such
actions on the 15 exotic 7-spheres mentioned above.