Related papers: Geometric structures on the quaternionic unit ball…

We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those special hypergeometric functions found by Heckman-Opdam, as well as the work of Couwenberg-Heckman-Looijenga on…

This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework…

This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to ${SO(4)}_{ir}\subset GL(8,\mathbb R)$, the image of an irreducible representation of $SO(4)$ on $\mathbb R^8$. We demonstrate that such a reduction can…

The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced…

We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted H\"older spaces.

Four dimensional simply connected Lie groups admitting a pseudo K\"ahler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,\omega)$ are parametrized up to complex isomorphism (where $J$ is a…

A 4-dimensional Riemannian manifold M, equipped with an additional tensor structure S, whose fourth power is minus identity, is considered. The structure S has a skew-circulant matrix with respect to some basis of the tangent space at a…

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric…

We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an $SU(3)\times Spin(6+n)$ structure within…

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such…

We define the operations of conformal change and elementary deformation in the setting of generalized complex geometry. Then we apply Swann's twist construction to generalized (almost) complex and Hermitian structures obtained by these…

Quasi-Hermitian quantum systems, including $\mathcal{PT}$-symmetric ones, can be mapped to equivalent Hermitian systems via a similarity transformation that redefines the inner product with a positive-definite metric operator. Although an…

Classical and quantum aspects of physical systems that can be described by Riemannian non degenerate superspaces are analyzed from the topological and geometrical points of view. For the N=1 case the simplest supermetric introduced in…

We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of…

This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that…

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We…

Special geometry is most known from 4-dimensional N=2 supergravity, though it contains also quaternionic and real geometries. In this review, we first repeat the connections between the various special geometries. Then the constructions are…

We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing…

We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…

This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…