Geodesic
Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs
Matematičeskie zametki, Tome 70 (2001) no. 5, pp. 718-735.

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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 $T^*P^n$. 
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     author = {V. V. Konnov},
     title = {Cotangent {Bundle} over {Projective} {Space} and the {Manifold} of {Nondegenerate} {Null-Pairs}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {718--735},
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     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_70_5_a6/}
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V. V. Konnov. Cotangent Bundle over Projective Space and the Manifold of Nondegenerate Null-Pairs. Matematičeskie zametki, Tome 70 (2001) no. 5, pp. 718-735. http://geodesic.mathdoc.fr/item/MZM_2001_70_5_a6/

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