Neifeld’s Connection Inducedon the Grassmann Manifold

Olga Belova

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2016)

  • Volume: 55, Issue: 1, page 11-14
  • ISSN: 0231-9721

Abstract

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The work concerns to investigations in the field of differential geometry. It is realized by a method of continuations and scopes of G. F. Laptev which generalizes a moving frame method and Cartan’s exterior forms method and depends on calculation of exterior differential forms. The Grassmann manifold (space of all m -planes) is considered in the n -dimensional projective space P n . Principal fiber bundle of tangent linear frames is arised above this manifold. Typical fiber of the principal fiber bundle is the linear group working in the tangent space to the Grassmann manifold. Neifeld’s connection is given in this fibering. It is proved by Cartan’s external forms method, that Bortolotti’s clothing of the Grassmann manifold induces this connection.

How to cite

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Belova, Olga. "Neifeld’s Connection Inducedon the Grassmann Manifold." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 55.1 (2016): 11-14. <http://eudml.org/doc/286713>.

@article{Belova2016,
abstract = {The work concerns to investigations in the field of differential geometry. It is realized by a method of continuations and scopes of G. F. Laptev which generalizes a moving frame method and Cartan’s exterior forms method and depends on calculation of exterior differential forms. The Grassmann manifold (space of all $m$-planes) is considered in the $n$-dimensional projective space $P_n$. Principal fiber bundle of tangent linear frames is arised above this manifold. Typical fiber of the principal fiber bundle is the linear group working in the tangent space to the Grassmann manifold. Neifeld’s connection is given in this fibering. It is proved by Cartan’s external forms method, that Bortolotti’s clothing of the Grassmann manifold induces this connection.},
author = {Belova, Olga},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Projective space; the Grassmann manifold; principal fiber bundle; Neifeld’s connection},
language = {eng},
number = {1},
pages = {11-14},
publisher = {Palacký University Olomouc},
title = {Neifeld’s Connection Inducedon the Grassmann Manifold},
url = {http://eudml.org/doc/286713},
volume = {55},
year = {2016},
}

TY - JOUR
AU - Belova, Olga
TI - Neifeld’s Connection Inducedon the Grassmann Manifold
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2016
PB - Palacký University Olomouc
VL - 55
IS - 1
SP - 11
EP - 14
AB - The work concerns to investigations in the field of differential geometry. It is realized by a method of continuations and scopes of G. F. Laptev which generalizes a moving frame method and Cartan’s exterior forms method and depends on calculation of exterior differential forms. The Grassmann manifold (space of all $m$-planes) is considered in the $n$-dimensional projective space $P_n$. Principal fiber bundle of tangent linear frames is arised above this manifold. Typical fiber of the principal fiber bundle is the linear group working in the tangent space to the Grassmann manifold. Neifeld’s connection is given in this fibering. It is proved by Cartan’s external forms method, that Bortolotti’s clothing of the Grassmann manifold induces this connection.
LA - eng
KW - Projective space; the Grassmann manifold; principal fiber bundle; Neifeld’s connection
UR - http://eudml.org/doc/286713
ER -

References

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  1. Belova, O., 10.1007/s10958-009-9649-y, . J. Math. Sci. 162, 5 (2009), 605–632. (2009) MR2475594DOI10.1007/s10958-009-9649-y
  2. Borisenko, A. A., Nikolaevskii, Yu. A., Grassmann’s manifolds and Grassmanian image of the submanifolds, . Succ. Math. Sci. 46, 2 (1991), 41–83 (in Russian). (1991) MR1125272
  3. Bortolotti, E., Connessioni nelle varieta luogo di spazi, . Rend. Semin. Fac. Sci. Univ. Cagliari 3 (1933), 81–89. (1933) Zbl0007.36604
  4. Laptev, G. F., Differential geometry of the embeded manifolds, . Proc. Moscow Math. Soc. 2 (1953), 275–383 (in Russian). (1953) MR0057601
  5. Malakhaltsev, M. A., About internal geometry of Neifeld’s connection, . Proc. Moscow Math. Soc. 2 (1986), 67–69 (in Russian). (1986) MR0842329
  6. Neifeld, E. G., Affine connections on the normalized manifold of planes in the projective space, . Proc. Moscow Math. Soc. 11 (1976), 48–55 (in Russian). (1976) MR0487828
  7. Norden, A. P., The theory of compositions, . In: Problemy Geometrii. Itogi Nauki i Tekhniki 10, VINITI, Moscow, 1978, 117–145 (in Russian). (1978) MR0540265
  8. Norden, A. P., Projective metrics on Grassmann manifolds, . News of High Schools, Math. 11 (1981), 80–83 (in Russian). (1981) Zbl0498.53012MR0662349
  9. Shevchenko, Yu. I., Equipments of centreprojective manifolds, . Kaliningrad, 2000 (in Russian). (2000) 

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