Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Projective space; the Grassmann manifold; principal fiber bundle; Neifeld’s connection
Projective space; the Grassmann manifold; principal fiber bundle; Neifeld’s connection
Summary:
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.
References:
[1] Belova, O.: Connections in fiberings associated with the Grassmann manifold and the space of centred planes. J. Math. Sci. 162, 5 (2009), 605–632. DOI 10.1007/s10958-009-9649-y | MR 2475594
[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). MR 1125272
[3] Bortolotti, E.: Connessioni nelle varieta luogo di spazi. Rend. Semin. Fac. Sci. Univ. Cagliari 3 (1933), 81–89. Zbl 0007.36604
[4] Laptev, G. F.: Differential geometry of the embeded manifolds. Proc. Moscow Math. Soc. 2 (1953), 275–383 (in Russian). MR 0057601
[5] Malakhaltsev, M. A.: About internal geometry of Neifeld’s connection. Proc. Moscow Math. Soc. 2 (1986), 67–69 (in Russian). MR 0842329
[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). MR 0487828
[7] Norden, A. P.: The theory of compositions. In: Problemy Geometrii. Itogi Nauki i Tekhniki 10, VINITI, Moscow, 1978, 117–145 (in Russian). MR 0540265
[8] Norden, A. P.: Projective metrics on Grassmann manifolds. News of High Schools, Math. 11 (1981), 80–83 (in Russian). MR 0662349 | Zbl 0498.53012
[9] Shevchenko, Yu. I.: Equipments of centreprojective manifolds. Kaliningrad, 2000 (in Russian).
Search
Partner of
