Curvature of the second kind and a conjecture of Nishikawa

In this paper we investigate manifolds for which the curvature operator of the second kind (following the terminology of Nishikawa (1986)) satisfies certain positivity conditions. Our main result settles Nishikawa’s conjecture that a manifold with positive curvature operator of the second kind is diffeomorphic to a sphere, by showing that such manifolds satisfy Brendle’s PIC1 condition. In dimension four we show that curvature of the second kind has a canonical normal form, and use this to classify Einstein four-manifolds for which the curvature (operator) of the second kind is five-non-negative. We also calculate the normal form for some explicit examples in order to show that this assumption is sharp.