Simeon Zamkovoy

Hyper-parahermitian manifolds with torsion

Abstract Necessary and sufficient conditions for the existence of a hyper-parahermitian connection with totally skew-symmetric torsion (HPKT-structure) are presented. It is shown that any HPKT-structure is locally generated by a real (potential) function. An invariant first order differential operator is defined on any almost hyper-paracomplex manifold showing that it is two-step nilpotent exactly when the almost hyper-paracomplex structure is integrable. A local HPKT-potential is expressed in terms of this operator. Examples of (locally) invariant HPKT-structures with closed as well as non-closed torsion 3-form on a class of (locally) homogeneous hyper-paracomplex manifolds (some of them compact) are constructed.

Geometry of paraquaternionic Kähler manifolds with torsion

Abstract We study the geometry of PQKT connections. We find conditions for the existence of a PQKT connection and prove that if it exists it is unique. We show that PQKT geometry persists in a conformal class of metrics.

Twistor and reflector spaces of almost para-quaternionic manifolds

The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds.We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.Under the action of the c-map, special K¨ahler manifolds are mapped into a class of quaternion-K¨ahler spaces. We explicitly construct the corresponding Swann bundle or hyperk¨ahler cone, and determine the hyperk¨ahler potential in terms of the prepotential of the special K¨ahler geometry.We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexion. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kaehler geometry, but from the perspective of non-integrable holonomy systems.An almost para-CR structure on a manifold

The decomposition of almost paracontact metric manifolds in eleven classes revisited

M

Canonical connections on paracontact manifolds

is given by a distribution

ParaHermitian and paraquaternionic manifolds

HM \subset TM

Conformal paracontact curvature and the local flatness theorem

together with a field

Para-Hermitian and Para-Quaternionic manifolds

K \in \Gamma({\rm End}(HM))

Non-existence of flat paracontact metric structures in dimension greater than or equal to five

of involutive endomorphisms of

Almost Paracontact Manifolds

HM