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

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
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
\Bibitem{Kon01} \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 \mathnet{http://mi.mathnet.ru/mz783} \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} 

• http://mi.mathnet.ru/eng/mz783
• https://doi.org/10.4213/mzm783
• http://mi.mathnet.ru/eng/mz/v70/i5/p718

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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
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
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