Geodesic
Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 1, pp. 111-120 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The second order transverse bundle $T^2_{}M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra $\mathbb {D}^2$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $\mathbb {D}^2$-smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $\mathbb {D}^2$-smooth foliated diffeomorphism between two second order transverse bundles maps the lift of a foliated linear connection into the lift of a foliated linear connection.
The second order transverse bundle $T^2_{}M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra $\mathbb {D}^2$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $\mathbb {D}^2$-smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $\mathbb {D}^2$-smooth foliated diffeomorphism between two second order transverse bundles maps the lift of a foliated linear connection into the lift of a foliated linear connection.
Classification : 53C12, 53C15, 58A20, 58A32
Keywords: Foliation; transverse bundle; second order transverse bundle; projectable linear connection; Lie derivative; Weil bundle 
@article{AUPO_2016_55_1_a12,
     author = {Shurygin, Vadim V. and Zubkova, Svetlana K.},
     title = {Lifts of {Foliated} {Linear} {Connectionsto} the {Second} {Order} {Transverse} {Bundles}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {111--120},
     year = {2016},
     volume = {55},
     number = {1},
     mrnumber = {3674605},
     zbl = {1365.58003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2016_55_1_a12/}
}
TY  - JOUR
AU  - Shurygin, Vadim V.
AU  - Zubkova, Svetlana K.
TI  - Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2016
SP  - 111
EP  - 120
VL  - 55
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AUPO_2016_55_1_a12/
LA  - en
ID  - AUPO_2016_55_1_a12
ER  - 
%0 Journal Article
%A Shurygin, Vadim V.
%A Zubkova, Svetlana K.
%T Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2016
%P 111-120
%V 55
%N 1
%U http://geodesic.mathdoc.fr/item/AUPO_2016_55_1_a12/
%G en
%F AUPO_2016_55_1_a12
Shurygin, Vadim V.; Zubkova, Svetlana K. Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 1, pp. 111-120. http://geodesic.mathdoc.fr/item/AUPO_2016_55_1_a12/

[1] Evtushik, L. E., Lumiste, Yu. G., Ostianu, N. M., Shirokov, A. P.: Differential-geometric structures on manifolds. In: Problemy Geometrii. Itogi Nauki i Tekhniki 9, VINITI Akad. Nauk SSSR, Moscow, 1979, 5–246. | MR | Zbl

[2] Gainullin, F. R., Shurygin, V. V.: Holomorphic tensor fields and linear connections on a second order tangent bundle. Uchen. Zapiski Kazan. Univ. Ser. Fiz.-matem. Nauki 151, 1 (2009), 36–50. | Zbl

[3] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin, 1993. | MR

[4] Molino, P.: Riemannian Foliations. Birkhäuser, Boston, 1988. | MR | Zbl

[5] Morimoto, A.: Prolongation of connections to tangent bundles of higher order. Nagoya Math. J. 40 (1970), 99–120. | DOI | MR

[6] Morimoto, A.: Prolongation of connections to bundles of infinitely near points. J. Different. Geom. 11, 4 (1976), 479–498. | DOI | MR | Zbl

[7] Pogoda, Z.: Horizontal lifts and foliations. Rend. Circ. Mat. Palermo 38, 2, suppl. no. 21 (1989), 279–289. | MR | Zbl

[8] Shurygin, V. V.: Structure of smooth mappings over Weil algebras and the category of manifolds over algebras. Lobachevskii J. Math. 5 (1999), 29–55. | MR | Zbl

[9] Shurygin, V. V.: Smooth manifolds over local algebras and Weil Bundles. J. Math. Sci. 108, 2 (2002), 249–294. | DOI | MR | Zbl

[10] Shurygin, V. V.: Lie jets and symmetries of geometric objects. J. Math. Sci. 177, 5 (2011), 758–771. | DOI | MR

[11] Vishnevskii, V. V.: Integrable affinor structures and their plural interpretations. J. Math. Sci. 108, 2 (2002), 151–187. | DOI | MR

[12] Wolak, R.: Normal bundles of foliations of order $r$. Demonstratio Math. 18, 4 (1985), 977–994. | MR | Zbl