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The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle T ∗ M of a smooth manifold M, induced by a symmetric linear connection ∇ on M. In this paper we deal with a natural Riemann extension g ¯ , which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure J ¯ on the cotangent bundle T ∗ M of an almost complex manifold (M , J , ∇) with a symmetric linear connection ∇ such that (T ∗ M , J ¯ , g ¯) is an almost complex manifold, where the natural Riemann extension g ¯ is a Norden metric. We obtain necessary and sufficient conditions for (T ∗ M , J ¯ , g ¯) to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold (M , J , ∇ ′) endowed with an almost complex connection ∇ ′ ( ∇ ′ J = 0 ). We investigate the harmonicity with respect to g ¯ of the almost complex structure J ¯ , according to the type of the base manifold. Moreover, we define an almost hypercomplex structure (J ¯ 1 , J ¯ 2 , J ¯ 3) on the cotangent bundle T ∗ M 4 n of an almost hypercomplex manifold (M 4 n , J 1 , J 2 , J 3 , ∇) with a symmetric linear connection ∇. The natural Riemann extension g ¯ is a Hermitian metric with respect to J ¯ 1 and a Norden metric with respect to J ¯ 2 and J ¯ 3 . [ABSTRACT FROM AUTHOR]

An existence of invariant hypercomplex structure on compact homogeneous spaces implies strong restrictions on their root structure and consequently on their characteristic Pontrjagin classes and the corresponding Chern classes. We describe these constraints by making use of Lie theory. [ABSTRACT FROM AUTHOR]

We construct explicit examples of quaternion-Kähler and hypercomplex structures on bundles over hyperKähler manifolds. We study the infinitesimal symmetries of these examples and the associated Galicki–Lawson quaternion-Kähler moment map. By performing the QK reduction we produce several explicit QK metrics. Moreover we are led to a new proof of a hyperKähler/quaternion-Kähler type correspondence. We also give examples of other Einstein metrics and balanced Hermitian structures on these bundles. [ABSTRACT FROM AUTHOR]

We study 4n-dimensional smooth manifolds admitting a SO ∗ (2 n) - or a SO ∗ (2 n) Sp (1) -structure, where SO ∗ (2 n) is the quaternionic real form of SO (2 n , C) . We show that such G-structures, called almost hypercomplex/quaternionic skew-Hermitian structures, form the symplectic analogue of the better known almost hypercomplex/quaternionic-Hermitian structures (hH/qH for short). We present several equivalent definitions of SO ∗ (2 n) - and SO ∗ (2 n) Sp (1) -structures in terms of almost symplectic forms compatible with an almost hypercomplex/quaternionic structure, a quaternionic skew-Hermitian form, or a symmetric 4-tensor, the latter establishing the counterpart of the fundamental 4-form in almost hH/qH geometries. The intrinsic torsion of such structures is presented in terms of Salamon's E H -formalism, and the algebraic types of the corresponding geometries are classified. We construct explicit adapted connections to our G-structures and specify certain normalization conditions, under which these connections become minimal. Finally, we present the classification of symmetric spaces K/L with K semisimple admitting an invariant torsion-free SO ∗ (2 n) Sp (1) -structure. This paper is the first in a series aiming at the description of the differential geometry of SO ∗ (2 n) - and SO ∗ (2 n) Sp (1) -structures. [ABSTRACT FROM AUTHOR]

We study almost hypercomplex skew-Hermitian structures and almost quaternionic skew-Hermitian structures, as the geometric structures underlying SO ∗ (2 n) - and SO ∗ (2 n) Sp (1) -structures, respectively. The corresponding intrinsic torsions were computed in the previous article in this series, and the algebraic types of the geometries were derived, together with the minimal adapted connections (with respect to certain normalizations conditions). Here we use these results to present the related first-order integrability conditions in terms of the algebraic types and other constructions. In particular, we use distinguished connections to provide a more geometric interpretation of the presented integrability conditions and highlight some features of certain classes. The second main contribution of this note is the illustration of several specific types of such geometries via a variety of examples. We use the bundle of Weyl structures and describe examples of SO ∗ (2 n) Sp (1) -structures in terms of functorial constructions in the context of parabolic geometries. [ABSTRACT FROM AUTHOR]

In the paper, almost hypercomplex manifolds with Hermitian-Norden metrics of the lowest dimension are studied. The considered manifolds are constructed on 4-dimensional Lie groups. A relation between the classes of the classification of 4-dimensional indecomposable real Lie algebras and the classification of the manifolds under study is established. The basic geometrical characteristics of the constructed manifolds are studied in the frame of the mentioned classification of Lie algebras. [ABSTRACT FROM AUTHOR]

The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle T∗M of a smooth manifold M, induced by a symmetric linear connection ∇ on M. In this paper we deal with a natural Riemann extension g¯, which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure J¯ on the cotangent bundle T∗M of an almost complex manifold (M,J,∇) with a symmetric linear connection ∇ such that (T∗M,J¯,g¯) is an almost complex manifold, where the natural Riemann extension g¯ is a Norden metric. We obtain necessary and sufficient conditions for (T∗M,J¯,g¯) to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold (M,J,∇′) endowed with an almost complex connection ∇′ (∇′J=0). We investigate the harmonicity with respect to g¯ of the almost complex structure J¯, according to the type of the base manifold. Moreover, we define an almost hypercomplex structure (J¯1,J¯2,J¯3) on the cotangent bundle T∗M4n of an almost hypercomplex manifold (M4n,J1,J2,J3,∇) with a symmetric linear connection ∇. The natural Riemann extension g¯ is a Hermitian metric with respect to J¯1 and a Norden metric with respect to J¯2 and J¯3.