New York Journal of Mathematics
Volume 28 (2022), 1085-1098


Jeffrey S. Case

Global obstructions to conformally Einstein metrics in dimension six

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Published: July 27, 2022.
Keywords: Einstein metric, global conformal invariant, global obstruction.
Subject [2020]: Primary 5C25; Secondary 53C18.

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.


I thank Misha Gromov and Claude LeBrun for helpful comments and encouragement. I also thank the anonymous referee for many helpful comments. This work was partially supported by the Simons Foundation (Grant #524601).

Author information

Jeffrey S. Case:
Department of Mathematics
Penn State University
University Park, PA 16802, USA