Curvature of the Second kind and a conjecture of Nishikawa

TL;DR

This paper proves Nishikawa's conjecture that manifolds with positive curvature of the second kind are diffeomorphic to spheres, and classifies Einstein four-manifolds with specific curvature conditions, providing explicit normal forms and sharpness examples.

Contribution

It confirms Nishikawa's conjecture by linking positive second kind curvature to the PIC1 condition and classifies certain Einstein four-manifolds using normal forms.

Findings

Manifolds with positive second kind curvature are diffeomorphic to spheres.

In dimension four, curvature of the second kind has a canonical normal form.

Classified Einstein four-manifolds with five-non-negative curvature of the second kind.

Abstract

In this paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the curvature (operator) of the second kind are positive are 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.