# Connections in fiberings associated with the Grassman manifold and the space of centered planes

Article

First Online:

## Abstract

In the present paper we study connections in the fiberings associated with the Grassmann manifold and the space of the centered planes. The work is related to the studies in differential geometry. In the paper, we use the method of continuations and scopes of G. F. Laptev which generalizes the moving frame method and the exterior forms method of Cartan; the method depends on calculation of exterior differential forms. In the paper, we develop a new method of research in Grassman manifolds and some generalization of the method which includes the theory of the induced connections of the spaces of planes and centered planes in the *n*-dimensional projective space.

## Keywords

Manifold Covariant Derivative Centered Plane Deformation Tensor Connection Form
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 1.O. O. Belova, “Connection in the fibering associated with the Grassman manifold,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 31, Kaliningrad (2000), pp. 8–11.Google Scholar - 2.O. O. Belova, “Curvature object of connection in the fibering associated with the Grassman manifold,” in:
*Problems of the Mathematical and Physical Sciences*[in Russian], Kaliningrad (2000), pp. 19–22.Google Scholar - 3.O. O. Belova, “Connections of three types in the fibering over the space of centered planes,” in:
*Proc. Math. N. I. Lobachevsky Center*[in Russian], Vol. 12, Kazan (2001), pp. 23–24.Google Scholar - 4.O. O. Belova, “Connections of three types in the fibering associated with the Grassman manifold,” in:
*Problems of the Mathematical and Physical Sciences*[in Russian], Kaliningrad (2001), pp. 3–5.Google Scholar - 5.O. O. Belova, “Covariant differential of the clothing quasitensor on the Grassman manifold,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 32, Kaliningrad (2001), pp. 13–17.Google Scholar - 6.O. O. Belova, “Connections in the principal fiber bundle over the area of the projective space,” in:
*Proc. Math. N. I. Lobachevsky Center*[in Russian], Vol. 18, Kazan (2002), pp. 9–10.Google Scholar - 7.O. O. Belova, “Geometric connections in the space of centered planes,” in:
*Reports Int. Math. Sem.: To 140th Anniversary of David Hilbert from Königsberg and 25th Anniversary of Mathematical Faculty*[in Russian], Kaliningrad (2002), pp. 100–105.Google Scholar - 8.O. O. Belova, “Geometric interpretation of connection in the fibering over the space of centered planes,” in:
*Problems of the Mathematical and Physical Sciences*[in Russian], Kaliningrad (2002), pp. 27–28.Google Scholar - 9.O. O. Belova, “Interpretation of the first-type connection in the fibering over the Grassman manifold,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 33, Kaliningrad (2002), pp. 14–17.Google Scholar - 10.O. O. Belova, “Bundle of first-type connections induced by an analog of the Norden’s normalization of centered planes space,” in:
*Proc. Joined Int. Sci. Conf. “New Geometry of Nature,”*Kazan (2003), pp. 51–54.Google Scholar - 11.O. O. Belova, “Connections of three types in the principal fiber bundle over the area of the projective space,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 34, Kaliningrad (2003), pp. 21–26.Google Scholar - 12.O. O. Belova, “Geometric connections in the space of centered planes,” in:
*Proc. Math. N. I. Lobachevsky Center*[in Russian], Vol. 21, Kazan (2003), p. 81.Google Scholar - 13.O. O. Belova, “Parallel displacements in the first-type connection of the space of centered planes,” in:
*Int. Conf. in the Field of Geometry and Analysis*[in Russian], Penza (2003), pp. 3–5.Google Scholar - 14.O. O. Belova, “Torsion tensor of the group subconnection on the Grassman manifold,” in:
*Theses of Reports of Int. Conf. “Geometry in Odessa—2004. Dif. Geom. and Its Applications”*[in Russian], Odessa (2004), pp. 8–10.Google Scholar - 15.O. O. Belova, “Torsion quasitensor of the group subconnection in the space of centered planes,” in:
*Collection Trans. Int. Geom. Seminar by G. F. Laptev*[in Russian], Penza (2004), pp. 5–8.Google Scholar - 16.V. I. Bliznikas, “Lie’s nonholonomic differentiation and a linear connection in the space of basic elements,”
*Lithuanian Math. Collection*,**6**, No. 2, 141–209 (1966).MATHMathSciNetGoogle Scholar - 17.V. I. Bliznikas, “Some questions of geometry of hypercomplexes of lines,” in:
*Proc. Geom. Seminar*[in Russian], Vol. 6, Moscow (1974), pp. 43–111.Google Scholar - 18.I. V. Bliznikene, “About geometry of hemiholonomic congruence of the first type,” in:
*Proc. Geom. Seminar*[in Russian], Vol. 3, VINITI, Moscow (1971), pp. 125–148.Google Scholar - 19.A. A. Borisenko and Yu. A. Nikolaevskii, “Grassmann manifolds and Grassmanian image of submanifolds,”
*Usp. Mat. Nauk*,**46**, No. 2, 41–83 (1991).MathSciNetGoogle Scholar - 20.E. Bortolotti, “Connessioni nelle varieta luogo di spazi,”
*Rend. Sem. Fac. Sci. Univ. Cagliari*, No. 3, 81–89 (1933).Google Scholar - 21.L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential-geometrical structures on manifolds,” in:
*Problems of Geometry*, Vol. 9, VINITI, Moscow (1979), pp. 5–247.Google Scholar - 22.G. F. Laptev, “Differential geometry of the embedded manifolds,”
*Tr. Mosk. Mat. Obshch.*,**2**, 275–383 (1953).MATHMathSciNetGoogle Scholar - 23.G. F. Laptev and N. M. Ostianu, “Distributions of
*m*-dimensional linear elements in the space of projective connection. I,” in:*Proc. Geom. Seminar*[in Russian], Vol. 3, VINITI, Moscow (1971), pp. 49–94.Google Scholar - 24.Yu. G. Lumiste, “Induced connection in the embeded projective and affine fiberings,”
*Uchen. Zap. Tartu. Gos. Univ.*, No. 177, 6–42 (1965).Google Scholar - 25.E. G. Neifeld, “Normalized space of
*m*-planes of the*n*-dimensional projective space,” in:*Theses of Reports of the second All-Union Geom. Conf.*[in Russian], Kharkov (1964), pp. 190–191.Google Scholar - 26.E. G. Neifeld, “Affine connections on the normalized manifolds of planes of the projective space,”
*Izv. Vyssh. Uchebn. Zaved., Mat.*, No. 11, 48–55 (1976).Google Scholar - 27.A. P. Norden,
*Spaces with Affine Connection*[in Russian], Nauka, Moscow (1976).MATHGoogle Scholar - 28.K. V. Polyakova, “Interpretation of subtensors of the deformation tensor on a surface,” in:
*Problems of the Mathematical and Physical Sciences*[in Russian], Kaliningrad (2001), pp. 20–22.Google Scholar - 29.Yu. I. Shevchenko, “About clothings of manifolds of planes in the projective space,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 9, Kaliningrad (1978), pp. 124–133.Google Scholar - 30.Yu. I. Shevchenko, “Parallel transposition of a figure in a linear combination of connection,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 18, Kaliningrad (1987), pp. 115–120.Google Scholar - 31.Yu. I. Shevchenko, “Connection in compound manifold and its continuation,” in:
*Differential Geometry of Figure Manifolds*[in Russian], No. 23, Kaliningrad (1992), pp. 110–118.Google Scholar - 32.Yu. I. Shevchenko,
*Clothings of Holonomic and Nonholonomic Smooth Manifolds*[in Russian], Kaliningrad (1998).Google Scholar - 33.Yu. I. Shevchenko,
*Clothings of Centreprojective Manifolds*[in Russian], Kaliningrad (2000).Google Scholar - 34.Yu. I. Shinkunas, “On distribution of
*m*-dimensional planes in the*n*-dimensional Riemannian space,” in:*Proc. Geom. Seminar*[in Russian], Vol. 5, VINITI, Moscow (1974), pp. 123–133.Google Scholar - 35.A. V. Skryagina, “A bundle of first-type connections induced by the Bortolotti’s clothing of a plane surface,” in:
*Problems of the Mathematical and Physical Sciences*[in Russian], Kaliningrad (2000), pp. 35–38.Google Scholar - 36.V. V. Vagner, “Theory of a compound manifold,” in:
*Proc. Seminar on Vector and Tensor Analysis with Their Applications to Geometry, Mechanics, and Physics*[in Russian], No. 8, Moscow–Leningrad (1950), pp. 11–72.Google Scholar

## Copyright information

© Springer Science+Business Media, Inc. 2009