On the curvature operator in dimensions $4n$

Abstract

We study oriented Riemannian $4n$-manifolds whose Thorpe $2n^{\text{th}}$ curvature operator ${\hat{R}}_{2n}:\hat{I}{›}^{2n}⟶\hat{I}{›}^{2n}$, or its Weyl analogue ${\hat{W}}_{2n}$, commutes with the Hodge star. For pure curvature operators this commuting condition becomes a finite system of hafnian identities in the eigenvalues of the curvature operator, which we analyze in two subclasses, including the locally conformally flat case. We further observe that $\ast {\hat{W}}_{2n}={\hat{W}}_{2n}\ast$ is a new conformal invariant in dimensions $4n$, providing higher-dimensional analogues of self-duality. Finally, we give sufficient conditions ensuring nonnegativity of the Euler characteristic and relate these conditions to normal forms.

23 pages