Journal
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS
Volume 82, Issue -, Pages -Publisher
ELSEVIER
DOI: 10.1016/j.difgeo.2022.101872
Keywords
Conformal invariant; Conformally invariant one-form
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This article introduces a natural conformally invariant form that is closely related to the Pfaffian of the Weyl tensor and top degree Pontrjagin forms. It discusses the properties and functions of these forms on manifolds.
We construct a natural conformally invariant one-form of weight -2k on any 2k dimensional pseudo-Riemannian manifold which is closely related to the Pfaffian of the Weyl tensor. On oriented manifolds, we also construct natural conformally invariant one-forms of weight -4k on any 4k-dimensional pseudo-Riemannian manifold which are closely related to top degree Pontrjagin forms. The weight of these forms implies that they define functionals on the space of conformal Killing fields. On Riemannian manifolds, we show that this functional is trivial for the former form but not for the latter forms. As a consequence, we obtain global obstructions to the existence of an Einstein metric in a given conformal class.(c) 2022 Published by Elsevier B.V.
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