Geometric structures of vectorial type

Abstract

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.

11 pages, Latex2e

References in corpus (2)

• On types of non-integrable geometries
• Almost Hermitian 6-Manifolds Revisited

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• Isometries and curvatures of tangent sphere bundles