Related papers: A new construction of homogeneous quaternionic man…

We construct a new family of curvature homogeneous pseudo-Riemannian manifolds modeled on $\mathbb{R}^{3k+2}$ for integers $k \geq 1$. In contrast to previously known examples, the signature may be chosen to be $(k+1+a, k+1+b)$ where $a,b…

The classification of homogeneous quaternionic manifolds has been done by Alekseevskii, Wolf et al using transitive solvable group of isometries. These manifolds are not generically symmetric, but there is a subset of quaternionic manifolds…

Classification results are given for (i) compact quaternionic K\"ahler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperK\"ahler manifolds with a cohomogeneity-two action of a semi-simple group…

Totally complex submanifolds of a quaternionic K\"{a}hler manifold are analogous to complex submanifolds of a K\"{a}hler manifold. In this paper, we construct an example of a non-compact totally complex submanifold of maximal dimension of a…

We analyze degenerate homogeneous structures of linear type in the pseudo-K\"ahler and para-K\"ahler cases. The local form and the holonomy of pseudo-K\"ahler or para-K\"ahler manifolds admitting such structure are obtained. In addition the…

We provide a complete classification of quaternionic skew-Hermitian symmetric spaces, namely symmetric spaces that admit a torsion-free ${\rm SO}^{*}(2n){\rm Sp}(1)$-structure for arbitrary $n>1$. Moreover, we prove that any homogeneous…

We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between…

In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic…

We give a procedure for constructing an $8n$-dimensional HKT Lie algebra starting from a $4n$-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K\"ahler, balanced) condition is…

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply…

Let $M$ be a pseudo-Riemannian spin manifold of dimension $n$ and signature $s$ and denote by $N$ the rank of the real spinor bundle. We prove that $M$ is locally homogeneous if it admits more than ${3/4}N$ independent Killing spinors with…

For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants…

We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. `conformal hypercomplex' manifolds) and quaternionic manifolds of 1 dimension less. This map is related to a method for constructing…

The holonomy group $G$ of a pseudo-quaternionic-K\"ahlerian manifold of signature $(4r,4s)$ with non-zero scalar curvature is contained in $\Sp(1)\cdot\Sp(r,s)$ and it contains $\Sp(1)$. It is proved that either $G$ is irreducible, or $s=r$…

Clifford geometric algebras of multivectors are introduced which exhibit a bilinear form which is not necessarily symmetric. Looking at a subset of bi-vectors in CL(K^{2n},B), we proof that theses elements generate the Hecke algebra…

We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic K\"ahler symmetric space if the dimension is at least 16. In…

We prove the existence of compact pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric, and represent all indefinite metric signatures in all dimensions $\,n\ge5$. Until now such…

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space (other than the octonion hyperbolic plane), and consider the space L(M) of oriented geodesics of M. The space L(M) is…

A symmetric quiver $(Q,\sigma)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on a…

A 2-form on a quaternionic-Kahler manifold (M, g) is called compatible (with the quaternionic structure) if it is a section of the direct sum bundle S^2(H) \oplus S^2(E). We construct a connection D on S^2(H) \oplus S^2(E)\oplus TM, which…