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

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 $T^*P^n$.