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

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 Kahlerian 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 \(\sigma :\mathfrak{N} \to T^* P^n\) is a symplectomorphism of the natural symplectic structure on \(\mathfrak{N}\) to the canonical symplectic structure on \(T^* P^n\).