Giulia Dileo

3-Quasi-Sasakian manifolds

In the present article we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular, we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.

THE GEOMETRY OF 3-QUASI-SASAKIAN MANIFOLDS

We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of this result both for bi-Legendrian and for paracontact manifolds. Finally new classes of examples of paracontact manifolds are presented.3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which appears to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kahler and quaternionic-Kahler geometries. Furthermore, we strongly improve the splitting results previously obtained; we prove that any 3-quasi-Sasakian manifold of rank 4l + 1 is 3-cosymplectic and any 3-quasi-Sasakian manifold of maximal rank is 3-α-Sasakian.

A CLASSIFICATION OF SPHERICAL SYMMETRIC CR MANIFOLDS

In this paper we classify the simply connected, spherical pseudohermitian manifolds whose Webster metric is CR-symmetric.

On nearly Sasakian and nearly cosymplectic manifolds

We prove that every nearly Sasakian manifold of dimension greater than five is Sasakian. This provides a new criterion for an almost contact metric manifold to be Sasakian. Moreover, we classify nearly cosymplectic manifolds of dimension greater than five.

Some Einstein nilmanifolds with skew torsion arising in CR geometry

We describe some new examples of nilmanifolds admitting an Einstein with skew torsion invariant Riemannian metric. These are affine CR quadrics, whose CR structure is preserved by the characteristic connection.

Generalized pseudohermitian manifolds

Abstract. We define a generalized pseudohermitian structure on an almost CR manifold as a pair (h,P), where h is a positive definite fiber metric h on compatible with J, and is a smooth projector such that . We show that to each generalized pseudohermitian structure one can associate a canonical linear connection on the holomorphic bundle which is invariant under equivalence. This fact allows us to solve the equivalence problem in the case where is a kind 2 distribution. We study the curvature of the canonical connection, especially for the classes of standard homogeneous manifolds and 3-Sasakian manifolds. The basic formulas for isopseudohermitian immersions are also obtained in the attempt to enlarge the theory of pseudohermitian immersions between strongly pseudoconvex pseudohermitian manifolds of hypersurface type.

A Note on 3‐Quasi‐Sasakian Geometry

3‐quasi‐Sasakian manifolds were recently studied by the authors as a suitable setting unifying 3‐Sasakian and 3‐cosymplectic geometries. In this paper some geometric properties of this class of almost 3‐contact metric manifolds are briefly reviewed, with an emphasis on those more related to physical applications.