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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
Full Text:
References:
 [1] M. BERGER , Les espaces symétriques non compacts (Ann. Éc. Norm., (3), Vol. 74:2, pp. 85-177, 1957 ). Numdam | MR 21 #3516 | Zbl 0093.35602 · Zbl 0093.35602 · numdam:ASENS_1957_3_74_2_85_0 · eudml:81724 [2] W. M. BOOTHBY and H. C. WANG , On contact manifolds (Ann. of Math., Vol. 68:3, pp. 721-734, 1958 ). MR 22 #3015 | Zbl 0084.39204 · Zbl 0084.39204 · doi:10.2307/1970165 [3] E. CARTAN , Leçons sur la géométrie des espaces de Riemann , Gauthier-Villars, Paris, 1951 . MR 13,491e | Zbl 0044.18401 · Zbl 0044.18401 · eudml:192543 [4] E. FALBEL and J. M. VELOSO , A parallelism for conformal sub-Riemannian geometry , Preprint, IMEUSP, 1993 . · Zbl 0854.53033 [5] E. FALBEL , J. M. VELOSO and J. A. VERDERESI , Constant curvature models in sub-Riemannian geometry (Matemática Contemporânea, Soc. Bras. Mat., Vol. 4, pp. 119-125, 1993 ). MR 95i:53034 | Zbl 0854.53033 · Zbl 0854.53033 [6] E. FALBEL , J. M. VELOSO and J. A. VERDERESI , The equivalence problem in sub-Riemannian geometry , Preprint, IMEUSP, 1993 . · Zbl 0854.53033 [7] S. HELGASON , Differential Geometry, Lie Groups, and Symmetric Spaces , Academic Press, 1978 . MR 80k:53081 | Zbl 0451.53038 · Zbl 0451.53038 [8] S. KOBAYASHI and K. NOMIZU , Foundations of Differential Geometry , Wiley Interscience Publishers, 1963 - 1969 . Zbl 0119.37502 · Zbl 0119.37502 [9] S. KOH , On affine symmetric spaces (Trans. Amer. Math. Soc., Vol. 119, pp. 291-309, 1965 ). MR 32 #1643 | Zbl 0139.39502 · Zbl 0139.39502 · doi:10.2307/1994052 [10] R. S. STRICHARTZ , Sub-Riemannian geometry (J. Diff. Geometry, Vol. 24, pp. 221-263, 1986 ). MR 88b:53055 | Zbl 0609.53021 · Zbl 0609.53021 [11] R. S. STRICHARTZ , Corrections to ”Sub-Riemannian geometry” (J. Diff. Geometry, Vol. 30, pp. 595-596, 1989 ). MR 90f:53081 | Zbl 0609.53021 · Zbl 0609.53021 [12] V. S. VARADARAJAN , Lie Groups, Lie Algebras and Their Representations , Springer-Verlag, New York, 1984 . MR 85e:22001 | Zbl 0955.22500 · Zbl 0955.22500 [13] S. M. WEBSTER , Pseudo-Hermitian structures on a real hypersurface (J. Diff. Geometry, Vol. 13, pp. 25-41, 1978 ). MR 80e:32015 | Zbl 0379.53016 · Zbl 0379.53016 [14] J. A. WOLF , Spaces of Constant Curvature , Publish or Perish, Boston, 1974 . MR 49 #7958 | Zbl 0281.53034 · Zbl 0281.53034
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