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On contact sub-Riemannian symmetric spaces. (English) Zbl 0836.53032
A contact sub-Riemannian manifold is a manifold equipped with a metric defined on a smooth subbundle \(\mathcal D\) which is the kernel of a contact form \(\vartheta\). Then the characteristic vector field \(\xi\) of \(d \vartheta\) ensures a natural Riemannian metric and a certain adapted connection \(\nabla\). If, moreover, there exists an involutive isometry of the space for every point which is a central symmetry when restricted to \(\mathcal D\), we speak of a sub-symmetric space.
Simulating Cartan’s classical methods, the authors characterize local sub-symmetry in terms of parallelism of curvature and torsion of \(\nabla\) along the subbundle \(\mathcal D\). Moreover, the structure of sub-symmetric spaces is linearized by means of special class of sub-orthogonal involutive Lie algebras which yields a decomposition theorem and complete classification of irreducible simply connected sub-symmetric spaces: every such a space is a homogeneous manifold fibered over a Hermitian symmetric space with fibers diffeomorphic to a circle and generated by the flow of a vector field \(\xi\); then \(\mathcal D\) is uniquely determined and the sub-metric is the pull-back of the metric on the base with two exceptions.
Reviewer: J.Chrastina (Brno)

MSC:
53C35 Differential geometry of symmetric spaces
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