Exotic Holonomy on Moduli Spaces of Rational Curves

Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces \(\Y\) of rational contact curves in a contact threefold \(\W\). Therefore, they are nat...

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Bibliographic Details
Published in:arXiv.org 1995-01
Main Authors: Quo-Shin, Chi, Schwachhöfer, Lorenz J
Format: Article
Language:English
Subjects:
Torsion
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Summary:Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces \(\Y\) of rational contact curves in a contact threefold \(\W\). Therefore, they are naturally contained in the moduli space \(\Z\) of all rational curves in \(\W\). We construct a connection on \(\Z\) whose restriction to \(\Y\) is torsion free. However, the connection on \(\Z\) has torsion unless both \(\Y\) and \(\Z\) are flat. We also show the existence of a new exotic holonomy which is a certain sixdimensional representation of \(\Sl \times \Sl\). We show that every regular \(H_3\)-connection (cf. \cite{Br}) is the restriction of a unique connection with this holonomy.
ISSN:2331-8422