Related papers: A class of Kaehler Einstein structures on the cota…

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex…

We review basic facts on the structure of nearly K\"ahler manifolds, focussing in particular on the six-dimensional case. A self-contained proof that nearly K\"ahler six-manifolds are Einstein is given by combining different known results.…

For an almost contact metric manifold $N$, we find conditions for which either the total space of an $S^1$-bundle over $N$ or the Riemannian cone over $N$ admits a strong K\"ahler with torsion (SKT) structure. In this way we construct new…

This articles is devoted to a description of the second-order differential geometry of torsion-free almost quaternionic skew-Hermitian manifolds, that is, of quaternionic skew-Hermitian manifolds $(M, Q, \omega)$. We provide a curvature…

Let $G/K$ be an irreducible Hermitian symmetric spaces of compact type with the standard homogeneous complex structure. Then the real symplectic manifold $(T^*(G/K),\Omega)$ has the natural complex structure $J^-$. We construct all…

Any quaternionic K\"ahler manifold $(\bar N,g_{\bar N},\mathcal Q)$ equipped with a Killing vector field $X$ with nowhere vanishing quaternionic moment map carries an integrable almost complex structure $J_1$ that is a section of the…

It is a classical fact that the cotangent bundle $T^* \M$ of a differentiable manifold $\M$ enjoys a canonical symplectic form $\Omega^*$. If $(\M,\j,g,\omega)$ is a pseudo-K\"ahler or para-K\"ahler $2n$-dimensional manifold, we prove that…

In earlier work we have shown that for certain geometric structures on a smooth manifold $M$ of dimension $n$, one obtains an almost para-K\"ahler--Einstein metric on a manifold $A$ of dimension $2n$ associated to the structure on $M$. The…

Let $M$ be a smooth manifold, let $TM$ be its tangent bundle and $T^{*}M$ its cotangent bundle. This paper investigates integrability conditions for generalized metrics, generalized almost para-complex structures, and generalized Hermitian…

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla $ which induces on the cotangent bundle $T^*M$ of $M$ a semi-Riemannian metric $\overline g$ with a neutral signature. The metric $\overline g$ is…

A para-K\"ahler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $ K^2 = \mathrm{Id} $ or, equivalently,…

We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition;…

For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a…

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics…

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic…

We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence K\"ahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant…

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E…

The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.

Given a real-analytic Riemannian manifold M there exists a canonical complex structure on part of its tangent bundle which turns leaves of the Riemannian foliation on TM into holomorphic curves. A Grauert tube over M of radius r, denoted as…

A compact symplectic manifold $(M, \omega)$ is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for $(M, \omega)$. This loosely means that there is a notion of harmonicity of differential…