Hermitean quadrics as contact manifolds
As it is well known the projective cotangent bundle of real projective space, a real contact manifold, is a real form of the projective cotangent bundle of complex projective space, a complex contact manifold. Hermitean quadrics (which are real manifolds in complex projective space) are real contact manifolds and are also real forms of the projective cotangent bundle of complex projective space. The latter is not evident and it is the purpose of the present paper to establish these assertions, exhibit their connection with the anti-polarities of classical projective geometry, and to show that these two types of real contact manifolds constitute all of the real forms of the projective cotangent bundle of complex projective space, as homogeneous contact manifolds. The development to these facts leads to the remark that Hermitean quadrics are principal circle bundles over products of complex projective spaces and generalize the Hopf bundle as real contact manifolds.
scientific article; zbMATH DE number 3884904