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Differential Geometry of Manifolds

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Belova O. O.

Centered planes in the projective connection space

DOI
10.5922/0321-4796-2020-51-4
Pages
29-38
projective connection space space of centered planes fi­be­ring connection

Abstract

The space  of centered planes is considered in the Cartan projec­ti­ve connection space . The space  is important because it has con­nec­tion with the Grassmann manifold, which plays an important role in geometry and topology, since it is the basic space of a universal vector bundle.
The space  is an n-dimensional differentiable manifold  with each point of which an n-dimensional projective space  containing this point is associated. Thus, the manifold  is the base, and the space  is the n-dimensional fiber “glued” to the points of the base.
A projective space  is a quotient space  of a linear space  with respect to the equivalence (collinearity) of non-zero vectors, that is . The projective space  is a manifold of di­men­sion n.
In this paper we use the Laptev — Lumiste invariant analytical meth­od of differential geometric studies of the space  of centered planes and introduce a fundamental-group connection in the associated bundle . The bundle  contains four quotient bundles. It is show that the connection object  is a quasi-tensor containing four subquasi-tensors that define connections in the corresponding quotient bundles.

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