Title: Geometry of complex manifolds similar to the Calabi-Eckmann manifolds
Abstract:In [4] Calabi and Eckmann showed that the product of two odd-dimensional spheres S 2p+1 x S 2q+1 (p,q > 1) is a complex manifold.As S 2p+1 x S 2q+1 is not Kaehlerian, the fundamental 2-form Ω of the H...In [4] Calabi and Eckmann showed that the product of two odd-dimensional spheres S 2p+1 x S 2q+1 (p,q > 1) is a complex manifold.As S 2p+1 x S 2q+1 is not Kaehlerian, the fundamental 2-form Ω of the Hermitian structure is not closed.However, dΩ does have a special form on S 2p+1 X S 2q+1 in fact, S 2p+1 x S 2q+1 admits two nonvanishing vector fields which are both Killing and analytic, and whose covariant forms determine Ω.Our purpose here is to study complex manifolds whose complex structures are similar to the complex structure on S 2p+1 X S 2q+1 .In § 1 we review the geometry of the Calabi-Eckmann manifolds.In § 2 we give some elementary properties of vector fields on a Hermitian manifold, and introduce the notion of a holomorphic pair of automorphisms and of a bicontact manifold.§ 3 continues the author's paper [2] on the differential geometry of principal toroidal bundles for the present case.In § 4 we discuss bicontact manifolds and, in particular, the integrable distributions of a bicontact structure on a Hermitian manifold.Finally in § 5 we give some results on the curvatures of a Hermitian manifold admitting a holomorphic pair of automorphisms.Read More