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Holonomie des connexions sans torsion

Abstract : We study the finite-dimensional Lie algebra representations in connection with their lattice of subrepresentations. We consider the case where the representation admits two pairs of supplementary invariant subspaces. We show that in this case the representation admits a canonical decomposition in three subrepresentations with well defined caracteristics. We strengthen the results for the situation where the representation admits a pair of supplementary invariant subspaces and an invariant reflexive form, respectively an invariant metric. In the geometric part we apply the preceding results to the study of the holonomy representations of a manifold equipped with a torsion-free connection or in particular a pseudo-Riemannian manifold. Finally we have a closer look at representations of type direct sum of V and V dual or V tensor the trivial representation of dimension 2 , which appear in this context, and we caracterize the torsion-free connections admitting a holonomy representation of this kind.
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Submitted on : Thursday, March 29, 2018 - 4:48:22 PM
Last modification on : Saturday, October 16, 2021 - 11:18:03 AM

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  • HAL Id : tel-01753764, version 1



Thomas Krantz. Holonomie des connexions sans torsion. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2007. Français. ⟨NNT : 2007NAN10030⟩. ⟨tel-01753764⟩



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