Almost positive curvature on the Gromoll-Meyer sphere

Abstract

Gromoll and Meyer have represented a certain exotic 7-sphere ${\Sigma }^7$ as a biquotient of the Lie group $G=Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on ${\Sigma }^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\ge 1$ which we describe explicitly.

8 pages, 1 figure, to appear in Proc. Amer. Math. Soc