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Horizontal lifts and foliations. (English) Zbl 0678.57013
Proc. Winter Sch. Geom. Phys., Srní/Czech. 1988, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 279-289 (1989).
Summary: [For the entire collection see Zbl 0672.00006.]
The purpose of this paper is to adapt the concept of a horizontal lift of a connection to a natural vector bundle to foliated manifolds. First, we recall the basic properties of transverse natural bundles. Second, we define the horizontal lift of basic connections of order r to a transverse natural vector bundle and we study its properties. Next we prove that for any basic connection of order r on a foliated manifold M there is exactly one horizontal lift to the transverse natural vector bundle which fulfills the satisfactory conditions. The results presented here generalize the results obtained by K. Yano and S. Ishihara [J. Math. Mech. 16, 1015-1029 (1967; Zbl 0152.204)], K. Yano and E.M. Patterson [J. Math. Soc. Jap. 19, 185-198 (1967; Zbl 0171.207)], J. Gancarzewicz [Differential geometry, Proc. 5th Int. Colloq., Santiago de Compostela/Spain 1984, Res. Notes Math. 131, 318-341 (1985; Zbl 0646.53028)], and J. Gancarzewicz and N. Rahmani [Ann. Pol. Math. 48, No.3, 291-295 (1988; Zbl 0667.53014)].

MSC:
57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
53C05 Connections (general theory)