Homogeneous metrics and spinors

Abstract: On a Riemannian manifold, the existence of a Killing spinor forces the metric to be Einstein. The homogeneous picture is completely understood; for negative scalar curvature only hyperbolic space occurs, whilst positive scalar curvature contains several examples, in particular the flag manifold $U(3)/T^3$ and the Sasaki quotients $U(n)/T^{n-1}$, themselves fibering over a flag manifold. For $n>3$, the quotient $U(n)/T^n$ has no Killing spinors, but the Dirac operator for the normal metric acts in a remarkably simple way on the space of invariant spinors, which can be described in a purely combinatorial way.

On pseudo-Riemannian manifolds, more homogeneous metrics with a Killing spinor appear. Through the lens of the Böhm-Lafuente theorem and the Heber-Lauret theory of standard solvmanifolds, this greater flexibility appears as a consequence of the failure of the Iwasawa condition, that lies at the heart of the correspondence between Einstein solvmanifolds and nilsolitons in positive-definite signature. I will show how imposing this condition recovers the same rigidity, and, by contrast, a construction that produces Einstein solvmanifolds not extending a nilsoliton that carry Killing spinors.

The talk will touch joint projects with F. A. Rossi and R. Segnan and with S. Salamon.

differential geometry

Audience: researchers in the topic

Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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