The category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.
@article{bwmeta1.element.doi-10_2478_s11533-013-0307-8,
author = {Nina Zhukova and Anna Dolgonosova},
title = {The automorphism groups of foliations with transverse linear connection},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {2076-2088},
zbl = {1288.53021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0307-8}
}
Nina Zhukova; Anna Dolgonosova. The automorphism groups of foliations with transverse linear connection. Open Mathematics, Tome 11 (2013) pp. 2076-2088. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0307-8/
[1] Bel’ko I.V., Affine transformations of a transversal projectable connection of a manifold with a foliation, Math. USSRSb., 1983, 45, 191–204 http://dx.doi.org/10.1070/SM1983v045n02ABEH001003
[2] Besse A.L., Einstein Manifolds, Classics Math., Springer, Berlin, 2008
[3] Kamber F.W., Tondeur P., G-foliations and their characteristic classes, Bull. Amer. Math. Soc., 1978, 84(6), 1086–1124 http://dx.doi.org/10.1090/S0002-9904-1978-14546-7 | Zbl 0405.57017
[4] Kobayashi S., Nomizu K., Foundations of Differential Geometry I, Interscience, New York-London, 1963 | Zbl 0119.37502
[5] Kriegl A., Michor P.W., Aspects of the theory of infinite-dimensional manifolds, Differential Geom. Appl., 1991, 1(2), 159–176 http://dx.doi.org/10.1016/0926-2245(91)90029-9
[6] Lewis A.D., Affine connections and distributions with applications to nonholonomic mechanics, In: Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics, Calgary, August 26–30, 1997, Rep. Math. Phys., 1998, 42(1-2), 135–164
[7] Macias-Virgós E., Sanmartín Carbón E., Manifolds of maps in Riemannian foliations, Geom. Dedicata, 2000, 79(2), 143–156 http://dx.doi.org/10.1023/A:1005217109018 | Zbl 0946.57033
[8] Michor P.W., Manifolds of Differentiable Mappings, Shiva Math. Ser., 3, Shiva, Nantwich, 1980
[9] Molino P., Propriétés cohomologiques et propriétés topologiques des feuilletages à connexion transverse projetable, Topology, 1973, 12, 317–325 http://dx.doi.org/10.1016/0040-9383(73)90026-8
[10] Molino P., Riemannian Foliations, Progr. Math., 73, Birkhäuser, Boston, 1988 http://dx.doi.org/10.1007/978-1-4684-8670-4
[11] Palais R.S., Foundations of Global Non-Linear Analysis, Benjamin, New York-Amsterdam, 1968 | Zbl 0164.11102
[12] Postnikov M.M., Lectures in Geometry V, Factorial, Moscow, 1998 (in Russian)
[13] Walker A.G., Connexions for parallel distributions in the large, Quart. J. Math. Oxford Ser., 1955, 6, 301–308 http://dx.doi.org/10.1093/qmath/6.1.301 | Zbl 0066.40203
[14] Willmore T.J., Connexions for systems of parallel distributions, Quart. J. Math. Oxford Ser., 1956, 7, 269–276 http://dx.doi.org/10.1093/qmath/7.1.269 | Zbl 0074.38001
[15] Zhukova N.I., Minimal sets of Cartan foliations, Proc. Steklov Inst. Math., 2007, 256(1), 105–135 http://dx.doi.org/10.1134/S0081543807010075 | Zbl 1246.37046
[16] Zhukova N.I., Global attractors of complete conformal foliations, Sb. Math., 2012, 203(3–4), 380–405 http://dx.doi.org/10.1070/SM2012v203n03ABEH004227 | Zbl 1307.53019