Geometric structures of vectorial type

Details Geometric structures of vectorial type Geometric structures of vectorial type

2005-09-07

arxiv.org/abs/math/0509147v1

Mathematics

Differential Geometry

11 pages, Latex2e

Scientific paper

10.1016/j.geomphys.2005.12.007

We study geometric structures of $\mathcal{W}_4$-type in the sense of A. Gray on a Riemannian manifold. If the structure group $\mathrm{G} \subset \SO(n)$ preserves a spinor or a non-degenerate differential form, its intrinsic torsion $\Gamma$ is a closed 1-form (Proposition \ref{dGamma} and Theorem \ref{Fixspinor}). Using a $\mathrm{G}$-invariant spinor we prove a splitting theorem (Proposition \ref{splitting}). The latter result generalizes and unifies a recent result obtained in \cite{Ivanov&Co}, where this splitting has been proved in dimensions $n=7,8$ only. Finally we investigate geometric structures of vectorial type and admitting a characteristic connection $\nabla^{\mathrm{c}}$. An interesting class of geometric structures generalizing Hopf structures are those with a $\nabla^{\mathrm{c}}$-parallel intrinsic torsion $\Gamma$. In this case, $\Gamma$ induces a Killing vector field (Proposition \ref{Killing}) and for some special structure groups it is even parallel.

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