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Mat. Zametki, 2001, Volume 70, Issue 5, Pages 718–735 (Mi mz783)  

This article is cited in 2 scientific papers (total in 2 papers)

Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs

V. V. Konnov

Samara State Teacher's Training University

Abstract: A nondegenerate null-pair of the real projective space $P^n$ consists of a point and of a hyperplane nonincident to this point. The manifold of all nondegenerate null-pairs $\mathfrak N$ carries a natural Kählerian structure of hyperbolic type and of constant nonzero holomorphic sectional curvature. In particular, $\mathfrak N$ is a symplectic manifold. We prove that $\mathfrak N$ is endowed with the structure of a fiber bundle over the projective space $P^n$, whose typical fiber is an affine space. The vector space associated to a fiber of the bundle is naturally isomorphic to the cotangent space to $P^n$. We also construct a global section of this bundle; this allows us to construct a diffeomorphism $\sigma$ between the manifold of nondegenerate null-pairs and the cotangent bundle over the projective space. The main statement of the paper asserts that the explicit diffeomorphism $\sigma\colon\mathfrak N\to T^*P^n$ is a symplectomorphism of the natural symplectic structure on $\mathfrak N$ to the canonical symplectic structure on $T^*P^n$.

DOI: https://doi.org/10.4213/mzm783

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English version:
Mathematical Notes, 2001, 70:5, 651–666

Bibliographic databases:

UDC: 514.76
Received: 22.03.2000
Revised: 26.09.2000

Citation: V. V. Konnov, “Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs”, Mat. Zametki, 70:5 (2001), 718–735; Math. Notes, 70:5 (2001), 651–666

Citation in format AMSBIB
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\by V.~V.~Konnov
\paper Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs
\jour Mat. Zametki
\yr 2001
\vol 70
\issue 5
\pages 718--735
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\crossref{https://doi.org/10.4213/mzm783}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1882345}
\zmath{https://zbmath.org/?q=an:1034.53077}
\transl
\jour Math. Notes
\yr 2001
\vol 70
\issue 5
\pages 651--666
\crossref{https://doi.org/10.1023/A:1012978927145}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000173100200007}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Konnov, “Kähler geometry of hyperbolic type on the manifold of nondegenerate $m$-pairs”, J. Math. Sci., 141:1 (2007), 1004–1015  mathnet  crossref  mathscinet  zmath
    2. Yu. L. Giluch, “A real analogue of the Bryant transformation and rational integral curves of a given distribution in $\mathbb P^3$”, Russian Math. (Iz. VUZ), 49:6 (2005), 72–77  mathnet  mathscinet  elib
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