On the curvature operator in dimensions $4n$

TL;DR

This paper investigates the properties of the curvature operator in 4n-dimensional Riemannian manifolds, introducing new invariants and conditions that influence topological characteristics like the Euler characteristic.

Contribution

It introduces a new conformal invariant in dimensions 4n and analyzes the eigenvalue identities of the curvature operator in specific subclasses.

Findings

Commutation with Hodge star leads to hafnian eigenvalue identities.

Identifies a new conformal invariant in 4n dimensions.

Provides conditions for nonnegativity of the Euler characteristic.

Abstract

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.