A positively curved metric on the Gromoll-Meyer sphere

RESUMO

We construct an explicit deformation of the bi-invariant metric on Sp(2) whose induced metrics on the Gromoll-Meyer exotic 7-sphere have positive sectional curvature for all sufficiently small positive parameters. The deformation is determined by a fixed self-adjoint operator compatible with the quotient action. The proof combines a uniform curvature estimate with an algebraic analysis of the horizontal commuting planes. On the critical set where the lower-order positive terms vanish, we establish an explicit positive lower bound for the cubic coefficient of the curvature numerator. Uniform control of the fourth-order remainder and a compactness argument then show that every sectional curvature is positive