Global obstructions to conformally Einstein metrics in dimension six

TL;DR

This paper introduces a global conformal invariant on closed six-manifolds that obstructs the existence of conformally Einstein metrics, establishing its uniqueness and implications for Einstein metrics.

Contribution

It defines a nontrivial, unique global conformal invariant in six dimensions that obstructs conformally Einstein metrics, expanding understanding of geometric obstructions.

Findings

The invariant is nontrivial and unique up to a constant.

It provides a diffeomorphism invariant obstruction to Einstein metrics.

Discusses invariants for manifolds with infinite fundamental group.

Abstract

We present a global conformal invariant on closed six-manifolds which obstructs the existence of a conformally Einstein metric. We show that this obstruction is nontrivial and, up to multiplication by a constant, is the unique such invariant. This also gives rise to a (possibly trivial) diffeomorphism invariant which obstructs the existence of an Einstein metric. We also discuss global conformal invariants which obstruct the existence of a conformally Einstein metric on closed six-manifolds with infinite fundamental group.