Hostname: page-component-784d4fb959-xqrh7 Total loading time: 0 Render date: 2025-07-15T15:28:32.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Lifts of some tensor fields and connections to product preserving functors

Published online by Cambridge University Press:  22 January 2016

Jacek Gancarzewicz ,
Włodzimierz Mikulski  and
Zdzisław Pogoda
Jacek Gancarzewicz
Affiliation:
Instytut Matematyki UJ, ul. Reymonta 4, 30-059 Kraków, Poland e-mail:(JG) gancarze@im.uj.edu.pl, (WM) mikluski@im.uj.edu.pl, (ZP) pogoda@im.um.edu.pl
Włodzimierz Mikulski
Affiliation:
Instytut Matematyki UJ, ul. Reymonta 4, 30-059 Kraków, Poland e-mail:(JG) gancarze@im.uj.edu.pl, (WM) mikluski@im.uj.edu.pl, (ZP) pogoda@im.um.edu.pl
Zdzisław Pogoda
Affiliation:
Instytut Matematyki UJ, ul. Reymonta 4, 30-059 Kraków, Poland e-mail:(JG) gancarze@im.uj.edu.pl, (WM) mikluski@im.uj.edu.pl, (ZP) pogoda@im.um.edu.pl
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we define some lifts of tensor fields of types (1, k) and (0, k) as well as connections to a product preserving functor . We study algebraic properties of introduced lifts and we apply these lifts to prolongation of geometric structures from a manifold M to (M). In particular cases of the tangent bundle of pr -velocities and the tangent bundle of infinitesimal near points our constructions contain all constructions due to Morimoto (see [20]-[23]). In the cases of the tangent bundle our definitions coincide with the definitions of Yano and Kobayashi (see [31]). To construct our lifts and to study its properties we use only general properties of product preserving functors. All lifts verify so-called the naturality condition. It means that for a smooth mapping φ:M → N and for two φ-related geometric objects defined on M and N its lifts to (M) and (N) respectively are (φ)-related. We explain later the term φ-related for considered geometric objects.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 135 , September 1994 , pp. 1 - 41
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

[1] Cordero, L. A., Dodson, C. T. J., de Leon, M., On the differential geometry of frame bundles, Kluwer, Dordrecht, 1989.Google Scholar
[2] Crittenden, R., Covariant differentiation, Quart. J. Math. Oxford, 13 (1963), 285–297.Google Scholar
[3] Eck, D. J., Product-preserving functors on smooth manifolds, J. Pure Appl. Algebra, 42(1986), 133–140.Google Scholar
[4] Gancarzewicz, J., Complete lifts of tensor fields of type (1, k) to natural bundles, Zeszyty Nauk. UJ, Kraków, 23 (1982), 43–79.Google Scholar
[5] Gancarzewicz, J., Differential Geometry (Polish), Monografie PWN, Warszawa 1987.Google Scholar
[6] Gancarzewicz, J., Prolongation of conjugate connections, Zeszyty Naukowe Politechniki Slaskiej, Gliwice, 64 (1990), 61–73.Google Scholar
[7] Gancarzewicz, J., Mahi, S., Géodésiques dans le fibre tangent d’ordre supérieur, Univer. Iagellonicae Acta Matematica, XXVIII (1991), 233–243.Google Scholar
[8] Gancarzewicz, J., Mikulski, W., Pogoda, Z., Natural bundles and natural liftings. Prolongations of geometric structures, Proc. Conf. on Diff. Geometry and Its Applications, Opava, August 24–28, 1992, Silesian University, Opava (1993), 281–320, Ed. by Kowalski, O. and Krupka, D..Google Scholar
[9] Gancarzewicz, J., Mikulski, W.,Pogoda, Z., Properties of product preserving functors, Proc. Winter School on Diff. Geometry and Its Applications, Zdikov, January 1993 (to appear).Google Scholar
[10] Godbillon, C., Géométrie différentielle et méchanique, analitique, Paris 1969.Google Scholar
[11] Kainz, G., Michor, P., Natural transformations in differential geometry, Czech. Math. J., 37(112), 1987, 584–607.Google Scholar
[12] Kobayashi, K. Nomizu, Foundations of differencial geometry, vol. 1, Intersciences, New York 1963.Google Scholar
[13] Kolář, I., On natural operators on vector fields, Ann. Global Analysis and Geometry, 6(2) (1988), 109–117.CrossRefGoogle Scholar
[14] Kolář, I., Michor, P. W., Slovak, J., Natural operations in differential geometry, Springer Verlag Berlin, 1993.Google Scholar
[15] Kolář, I., Slovák, J., On the geometric functors on manifolds, Suppl. ai Rendiconti del Cir. Math, di Palermo, 21 (1989), 223–233.Google Scholar
[16] Luciano, O. O., Categories of multiplicative functors and Morimoto’s conjecture, Preprint 46, Inst. Fourier, Laboratoire des Mathématiques, Grenoble, 1986.Google Scholar
[17] Mikulski, W., Natural transformations trasforming functions and vector fields to functions on some natural bundles, Math. Bohemica, 117 (1992), 217–223.Google Scholar
[18] Morimoto, A., Prolongations of geometric structures, Lect. Notes Math. Inst. Nagoya Univ. 1969.Google Scholar
[19] Morimoto, A., Prolongations of G-structures to tangent bundles, Nagoya Math. J., 32 (1968), 67–108.Google Scholar
[20] Morimoto, A., Liftings of some type of tensor fields and connections to tangent bundles of pr -velocities, Nagoya Math. J., 40 (1970), 13–31.Google Scholar
[21] Morimoto, A., Prolongations of connections to tangential fibre bundles of higher order, Nagoya Math. J., 40 (1970), 85–97.Google Scholar
[22] Morimoto, A., Liftings of tensor fields and connections to tangent bundles of higer order, Nagoya Math. J., 40 (1970), 99–120.Google Scholar
[23] Morimoto, A., Prolongations of connections to bundles of infinitely near points, J. Diff. Geom., 11 (1976), 476–498.Google Scholar
[24] Nijenhuis, A., Natural bundles and their general properties, Diff. Geom. in honor of K. Yano, Tokyo 1972, 317–334.Google Scholar
[25] Palais, R. and Terng, C.-L., Natural bundles have a finite order, Topology, 16 (1977), 271–277.Google Scholar
[26] Pogoda, Z., Liftings of connections with respect to A-functor, Demonstr. Math., (in press).Google Scholar
[27] Salvioli, S. E., Theory of geometric objects, J. Diff. Geom., 7 (1972), 257–278.Google Scholar
[28] Slovak, J., Prolongations of connections and sprays to Weil functors, Suppl. ai Reindicondi Circolo Math. Palermo, 14 (1987).Google Scholar
[29] Weil, A., Thèorie des points proches sur les variétés differentiables, Coloque Geom. Diff. Strasbourg 1953, 111–117.Google Scholar
[30] Yano, K. and Ishihara, S., Tangent and cotangent bundles, Marcel Dekker, INC., New York, 1973.Google Scholar
[31] Yano, K., Kobayashi, S., Prolongations of tensor fields and connections to tangent bundles I., J. Math. Soc. Japan, 18 (1966), 194–210.Google Scholar