Journal of Mathematical Sciences

, 177:753 | Cite as

Degeneration of plane affine Stolyarov connections

  • Yu. I. Shevchenko


Consider a distribution of a plane in a projective space. A way of defining a plane affine Stolyarov connection associated with this distribution is proposed. It is set by the field of a connection object consisting of a connection quasitensor and a linear connection object. The object of this generalized affine connection defines torsion and curvature objects. We show that these objects are tensors. Conditions under which a plane affine Stolyarov connection is torsion-free or curvature-free are described. It is proved that the generalized affine connection with the connection quasitensor is the generalized Kronecker symbol degenerated into a linear connection.


Manifold Structural Equation Projective Space Linear Connection Plane Distribution 
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  1. 1.
    O. O. Belova, “Connections in the bundles associated with the Grassman manifold and the space of centered planes,” J. Math. Sci., 162, No. 5, 605–632 (2009).MATHCrossRefGoogle Scholar
  2. 2.
    L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential-geometrical structures on manifolds,” Itogi Nauki Tekh. Ser. Probl. Geom. Tr. Geom. Sem., 9, 5–247 (1979).Google Scholar
  3. 3.
    Sh. Kobayashi, Groups of Transformations in Differential Geometry [Russian translation], Nauka, Moscow (1986).Google Scholar
  4. 4.
    G. F. Laptev, “Differential geometry of the embedded manifolds,” Tr. Mosk. Mat. Obshch., 2, 275–382 (1953).MathSciNetMATHGoogle Scholar
  5. 5.
    G. F. Laptev and N. M. Ostianu, “Distributions of m-dimensional linear elements in the space of projective connection. I,” Itogi Nauki Tekh. Ser. Probl. Geom. Tr. Geom. Sem., 3, 49–93 (1971).MathSciNetGoogle Scholar
  6. 6.
    O. M. Omelyan and Yu. I. Shevchenko, “Reductions of the center-projective connection object and of affine torsion tensor on planes distribution,” Mat. Zametki, 84, No. 1, 99–107 (2008).MathSciNetGoogle Scholar
  7. 7.
    K. V. Polyakova, “Parallel displacements on the surface of a projective space,” J. Math. Sci., 162, No. 5, 675–709 (2009).MATHCrossRefGoogle Scholar
  8. 8.
    Yu. I. Shevchenko, “Common fundamental-group connection from the point of view of the bundles,” Differ. Geom. Mnogoobr. Figur, No. 21, 100–105 (1990).Google Scholar
  9. 9.
    Yu. I. Shevchenko, “Laptev and Lumiste methods of the defining of connections in principal fiber bundles,” Differ. Geom. Mnogoobr. Figur, No. 37, 179–187 (2006).Google Scholar
  10. 10.
    Yu. I. Shevchenko, “Affine Stolyarov connection on the planes distribution in projective space,” in: Int. Conf. “Geometry in Astrakhan—2007,” Astrakhan (2007), pp. 65–67.Google Scholar
  11. 11.
    Yu. I. Shevchenko, “Degenerated plane affine Stolyarov connection,” in: Int. Conf. “Laptev Readinds—2009,” Tver (2009), p. 41.Google Scholar
  12. 12.
    Yu. I. Shevchenko, “Plane affine Stolyarov connection associated with distribution,” Differ. Geom. Mnogoobr. Figur, No. 40, 152–160 (2009).Google Scholar
  13. 13.
    A. V. Stolyarov, “Projectively differential geometry of regular hyperstrip distribution of m-dimensional linear elements,” Itogi Nauki Tekh. Ser. Probl. Geom. Tr. Geom. Sem., 7, 117–151 (1975).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.I. Kant State University of RussiaKaliningradRussia

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