We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M. We prove that for any foliation (M,F) there exists an open, not necessarily connected, saturated, and everywhere dense subset M0 of M and a manifold L0 such that the induced foliation (M0, FM0) is formed by the fibers of a locally trivial fibration with the standard fiber L0 over (possibly, non-Hausdorff) smooth q-dimensional manifold. In the case of codimension 1, the induced foliation on each connected component of the manifold M0 is formed by the fibers of a locally trivial fibration over a circle or over a line.
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References
K. Millett, “Generic properties of proper foliations,” Santa Barbara Univ. Math. J. 138, 131–138 (1984).
N. I. Zhukova, “Minimal sets of Cartan foliations” [in Russian], Tr. Mat. Inst. Steklova 256, 115-147 (2007); English transl.: Proc. Steklov Inst. Math. 256, 105-135 (2007).
R. A. Blumenthal and J. J. Hebda, “Ehresmann connections for foliations,” Indiana Univ. Math. J. 33, No 4, 597–611 (1984).
I. Tamura, Topology of Foliations: An Introduction, Am. Math. Soc., Providence, RI (1992).
N. I. Zhukova, “Complete foliations with transverse rigid geometries and their basic automorphisms” [in Russian], Vestn. Ross. Univ. Druzh. Nar., Ser. Mat. Inform. Fiz. No 2, 14–35 (2009) [electronic version only]
G. Glimm, “Locally compact transformation group,” Trans. Am. Math. Soc. 101, 124–138 (1961).
N. Zhukova, “On the stability of leaves of Riemannian foliations,” Ann. Global Anal. Geom. 5, No 3, 261–271 (1987).
N. I. Zhukova, “Local and global stability of compact leaves of foliations,” Zh. Mat. Fiz. Anal. Geom. 9, No 3. 400–420 (2013).
N. I. Zhukova, “Local and global stability of leaves of conformal foliations,” In: Foliations 2012, pp. 215–233, World Scientific, Singapore (2013).
R. A. Blumenthal, “Cartan submersions and Cartan foliations,” Ill. Math. J. 31, No 2, 327–343 (1987).
N. I. Zhukova, “Global attractors of complete conformal foliations” [in Russian] Mat. Sb. 203, No. 3, 79–106 (2012); English transl.: Sb. Math. 203, No. 3, 380–405 (2012).
Sh. Kobayashi and K. Nomizu, Foundation of Differential Geometry. I, John Wiley & Sons, New York etc. (1963).
Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer, Berlin (1995).
R. Hermann, “On differential geometry of foliations,” Ann. Math. 72, No 3, 445–457 (1960).
D. Epstein, K. Millett, and D. Tischler, “Leaves without holonomy,” J. London Math. Soc. 16, 548–552 (1977).
S. Kashiwabara, “The decomposition of differential manifolds and its applications,” Tohoku Math. J. 11, No 1, 43–53 (1959).
B. L. Reinhart, “Foliated manifolds with bundle-like metrics,” Ann. Math. 69, No. 1, 119–132 (1959).
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Translated from Problemy Matematicheskogo Analiza 85, June 2016, pp. 107-117.
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Zhukova, N.I. Typical Properties of Leaves of Cartan Foliations with Ehresmann Connection. J Math Sci 219, 112–124 (2016). https://doi.org/10.1007/s10958-016-3087-4
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DOI: https://doi.org/10.1007/s10958-016-3087-4