  We begin a systematic study of a curvature condition (strongly positive
curvature) which lies strictly between positive curvature operator and positive
sectional curvature, and stems from the work of Thorpe in the 1970s. We prove
that this condition is preserved under Riemannian submersions and Cheeger
deformations, and that most compact homogeneous spaces with positive sectional
curvature satisfy it.
