On different notions of positivity of curvature

We study interactions between the geometry and topology of Riemannian manifolds that satisfy curvature positivity conditions closely related to positive sectional curvature (sec>0). First, we discuss two notions of weakly positive curvature, defined in terms of averages of pairs of sectional curvatures. The manifold S^2 x S^2 is proved to satisfy these curvature positivity conditions, implying it satisfies a property intermediate between sec>0 and positive Ricci curvature, and between sec>0 and nonnegative sectional curvature. Combined with surgery techniques, this construction allows to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive. Second, we study the notion of strongly positive curvature, which is intermediate between sec>0 and positive-definiteness of the curvature operator. We elaborate on joint work with Mendes, which yields the classification of simply-connected homogeneous spaces that admit an invariant metric with strongly positive curvature. These methods are then used to study the moduli space of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds and on Berger spheres.