We study oriented Riemannian $4n$-manifolds whose Thorpe $2n^{ \text{ th}}$ curvature operator $\hat{ R}_{ 2n}\colon\Lambda^{ 2n} \longrightarrow \Lambda^{ 2n}$, 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 $*\hat{ W}_{ 2n} = \hat{ W}_{ 2n}*$ 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.
Categories: math.DG