an:00812648
Zbl 0836.53032
Falbel, Elisha; Gorodski, Claudio
On contact sub-Riemannian symmetric spaces
EN
Ann. Sci. ??c. Norm. Sup??r. (4) 28, No. 5, 571-589 (1995).
00029128
1995
j
53C35
Lie algebra; fibered space; sub-Riemannian manifold; sub-symmetric space; classification
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.
J.Chrastina (Brno)