On homogeneous connections with exotic holonomy |
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Authors: | Lorenz J Schwachhöfer |
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Institution: | (1) Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany;(2) Present address: Mathematisches Institut, Universität Leipzig, 04109 Leipzig, Germany |
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Abstract: | In Proc. Symp. Pure Math.
53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H
3-connections (G
3-connections resp.).In this paper, we give a complete classification of homogeneous G
3-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G
3-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G
3-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc.
345 (1994), 293–321) which states that there are no compact manifolds with an H
3-connection. |
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Keywords: | Primary: 53A15 Secondary: 53B05 |
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