Pseudo-Riemannian manifolds with recurrent spinor fields

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 connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.Comment: 13 pages, the final versio

Holonomy of Einstein Lorentzian manifolds

The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp., vacuum Einstein) Lorentzian manifold, the direct constructions are given. Also the holonomy algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The classification of the holonomy algebras of Lorentzian manifolds is reviewed and a complete description of the spaces of curvature tensors for these holonomies is given.Comment: Dedicated to to Mark Volfovich Losik on his 75th birthday. This version is an extended part of the previous version; another part of the previous version is extended and submitted as arXiv:1001.444

On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines

We study transformations of coordinates on a Lorentzian Einstein manifold with a parallel distribution of null lines and show that the general Walker coordinates can be simplified. In these coordinates, the full Lorentzian Einstein equation is reduced to equations on a family of Einstein Riemannian metrics.Comment: Dedicated to Dmitri Vladimirovich Alekseevsky on his 70th birthda

Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras \g\subset\osp(p,q|2m) of Riemannian supermanifolds under the assumption that \g is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.Comment: 27 pages, the final versio

Covariant derivative of the curvature tensor of pseudo-K\"ahlerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-K\"ahlerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-K\"ahlerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.Comment: the final version accepted to Annals of Global Analysis and Geometr

Pseudo-Riemannian VSI spaces

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of their polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the ${\bf S}_i$- and ${\bf N}$-properties, and show that if the curvature tensors of the space possess the ${\bf N}$-property then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null-congruence. We also discuss the related Walker metrics.Comment: 14 page

Photochromism and Electrochemistry of a Dithienylcyclopentene Electroactive Polymer

A bifunctional substituted dithienylcyclopentene photochromic switch bearing electropolymerisable methoxystyryl units, which enable immobilization of the photochromic unit on conducting substrates, is reported. The spectroscopic, electrochemical, and photochemical properties of a monomer in solution are compared with those of the polymer formed through oxidative electropolymerization. The electroactive polymer films prepared on gold, platinum, glassy carbon, and indium titanium oxide (ITO) electrodes were characterized by cyclic voltammetry, X-ray photoelectron spectroscopy (XPS), and atomic force microscopy (AFM). The thickness of the films formed is found to be limited to several monolayer equivalents. The photochromic properties and stability of the polymer films have been investigated by UV/vis spectroscopy, electrochemistry, and XPS. Although the films are electrochemically and photochemically stable, their mechanical stability with respect to adhesion to the electrode was found to be sensitive to both the solvent and the electrode material employed, with more apolar solvents, glassy carbon, and ITO electrodes providing good adhesion of the polymer film. The polymer film is formed consistently as a thin film and can be switched both optically and electrochemically between the open and closed state of the photochromic dithienylethene moiety.

The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds

In [B-I] the weakly irreducible, non-irreducible holonomy algebras of Lorentzian manifolds were divided into 4 types. We describe the structure of the spaces of curvature tensors for algebras of each type, this let us reduce the classification problem for the weakly irreducible, non-irreducible holonomy algebras of Lorentzian manifolds to the classification of irreducible subalgebras of so(k) that satisfy an algebraic criteria close to the Bianchi identity. We make the classification of admissible holonomy algebras for Lorentzian manifolds of dimensions ≤ 11