Advertisement

Mathematical Notes

, Volume 104, Issue 5–6, pp 789–798 | Cite as

The Grassmann-like Manifold of Centered Planes

  • O. O. BelovaEmail author
Article
  • 7 Downloads

Abstract

Connections associated with the Grassmann-like manifold of centered planes in the multidimensional projective space are studied. A geometric interpretation of these connections in terms of maps and translations of equipping planes is given. An intrinsic analog of Norden’s strong normalization of the manifold under consideration is constructed.

Keywords

Cartan’s exterior form method Grassmann manifold Norden’s normalization average connection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. P. Finikov, Cartan’s Method of Exterior Forms inDifferentialGeometry. The Theory of Compatibility of Systems of Total and Partial Differential Equations (GITTL, Moscow–Leningrad, 1948) [in Russian].Google Scholar
  2. 2.
    M. A. Akivis and B. A. Rozenfel’d, Élie Cartan (MTsNMO, Moscow, 2007) [in Russian].zbMATHGoogle Scholar
  3. 3.
    L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, Differential-geometric structures on manifolds, in Problems in Geometry, Vol. 9, Itogi Nauki i Tekhniki (VINITI, Moscow, 1979) [J. Soviet Math. 14 (6), 1573–1719 (1980)].zbMATHGoogle Scholar
  4. 4.
    Yu. I. Shevchenko, “Laptev’s and Lumiste’s tricks for specifying a connection in a principal bundle,” Differ. Geom.Mnogoobr. Figur 37, 179–187 (2006).Google Scholar
  5. 5.
    A. A. Borisenko and Yu. A. Nikolaevskii, “Grassmannmanifolds and theGrassmann image of submanifolds,” UspekhiMat. Nauk 46 (2 (278)), 41–83 (1991) [RussianMath. Surveys 46 (2), 45–94 (1991)].Google Scholar
  6. 6.
    Yu. G. Lumiste, “Induced connections in immersed projective and affine bundles,” in Works inMathematics and Mechanics, Vol. 177, Uchenye Zapiski Tartuskogo Universiteta (Tartuskii Gos. Univ., Tartu, 1965), pp. 6–41 [in Russian].Google Scholar
  7. 7.
    Shiing-Shen Chern, Complex Manifolds (Univ. of Chicago, Chicago, 1956; Inostrannaya Literatura, Moscow, 1961).zbMATHGoogle Scholar
  8. 8.
    A. Bichara and G. Tallini, “On a characterization of the Grassmann manifold representing the planes in a projective space,” in Combinatorial and Geometric Structures and Their Applications, Ann. Discrete Math. (North-Holland, Amsterdam, 1982), Vol. 63, pp. 129–149.MathSciNetzbMATHGoogle Scholar
  9. 9.
    R. Di Gennaro, E. Ferrara Dentice, and P. M. Lo Re, “On the Grassmann space representing the lines of an affine space,” DiscreteMath. 312 (3), 699–704 (2012).MathSciNetzbMATHGoogle Scholar
  10. 10.
    M. Harandi, R. Hartley, M. Salzmann, and J. Trumpf, “Dictionary learning on Grassmann manifolds,” in Algorithmic Advances in Riemannian Geometry and Applications (Springer, Cham, 2016), pp. 145–172.CrossRefGoogle Scholar
  11. 11.
    D. Baralic, “How to understand Grassmannians?,” The Teaching of Mathematics 14 (2), 147–157 (2011).Google Scholar
  12. 12.
    M. M. Postnikov, Lectures in Geometry, Semester II: Linear Algebra and Differential Geometry (Nauka, Moscow, 1979;Mir, Moscow, 1983).Google Scholar
  13. 13.
    G. F. Laptev, “Differential geometry of immersed manifolds. Group-theoreticmethod of differential-geometric research,” in Trudy Moskov. Mat. Obshch. (GITTL, Moscow, 1953), Vol. 2, pp. 275–382 [in Russian].Google Scholar
  14. 14.
    Yu. I. Shevchenko, Equipments of Central-Projective Manifolds (Kaliningrad. Gos. Univ., Kaliningrad, 2000) [in Russian].Google Scholar
  15. 15.
    O. O. Belova, “Connection of the second type in the bundle associated with the Grassmann-like manifold of centered planes,” Differ. Geom. Mnogoobr. Figur 38, 6–12 (2007).MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. P. Norden, Affine Connection Spaces (Nauka, Moscow, 1976) [in Russian].zbMATHGoogle Scholar
  17. 17.
    B. N. Shapukov, Problems on Lie Groups and Their Applications (Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2002) [in Russian].Google Scholar
  18. 18.
    Yu. I. Shevchenko, “Parallel translations on a surface,” Differ. Geom. Mnogoobr. Figur 10, 154–158 (1979).Google Scholar
  19. 19.
    K. V. Polyakova, “Parallel translations of directions along a surface in the projective space,” Differ. Geom. Mnogoobr. Figur 27, 63–70 (1996).zbMATHGoogle Scholar
  20. 20.
    A. V. Chakmazyan, “A connection in normal bundles of normalized submanifolds V m in P n,” in Problems in Geometry, Vol. 10, Itogi Nauki i Tekhniki (VINITI, Moscow, 1978), pp. 55–74 [J. Soviet Math. 14 (3), 1205–1216 (1980)].Google Scholar
  21. 21.
    G. F. Laptev and N. M. Ostianu, “Distributions of m-dimensional linear elements in projective connection space. I,” in Trudy Geom. Sem. (VINITI, Moscow, 1971), Vol. 3, pp. 49–94 [in Russian].zbMATHGoogle Scholar
  22. 22.
    O. O. Belova, “The curvature tensor of the connection in a bundle over the Grassmann-like manifold of centered planes,” Differ. Geom. Mnogoobr. Figur 40, 18–28 (2009).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Immanuel Kant Baltic Federal UniversityKaliningradRussia

Personalised recommendations