Connected subgroups of SO(2,n) acting irreducibly on R^{2,n}

We classify all connected subgroups of SO(2, n) that act irreducibly on R^{2, n

Ambient connections realising conformal Tractor holonomy

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus for a $C$-space this example leads to an ambient metric in the weaker sense of \v{C}ap and Gover, and for an Einstein space to a Ricci-flat ambient metric in the sense of Fefferman and Graham.Comment: 17 page

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

How to find the holonomy algebra of a Lorentzian manifold

Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de~Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra $\mathfrak{g}\subset\mathfrak{so}(1,n-1)$ of a locally indecomposable Lorentzian manifold $(M,g)$ of dimension $n$ is different from $\mathfrak{so}(1,n-1)$, then it is contained in the similitude algebra $\mathfrak{sim}(n-2)$. There are 4 types of such holonomy algebras. Criterion how to find the type of $\mathfrak{g}$ are given, and special geometric structures corresponding to each type are described. To each $\mathfrak{g}$ there is a canonically associated subalgebra $\mathfrak{h}\subset\mathfrak{so}(n-2)$. An algorithm how to find $\mathfrak{h}$ is provided.Comment: 15 pages; the final versio

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

Geometry and holonomy of indecomposable cones

We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non-irreducible cones. The latter admit a parallel distribution of null k-planes, and we study the cases k = 1 in detail. We give structure theorems about the base manifold and in the case when the base manifold is Lorentzian, we derive a description of the cone holonomy. This result is obtained by a computation of certain cocycles of indecomposable subalgebras in so(1,n - 1).Dmitri Alekseevsky, Vicente Cortés and Thomas Leistne

Ambient metrics for nn-dimensional pppp-waves

We provide an explicit formula for the Fefferman-Graham-ambient metric of an $n$-dimensional conformal $pp$-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional $pp$-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional $pp$-waves have vanishing critical $Q$-curvature.Comment: 17 pages, in v2 footnote and references added and typos corrected, in v3 remark in the Introduction about Brinkmann's results corrected and footnote adde

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

Control of positive and negative magnetoresistance in iron oxide−iron nanocomposite thin films for tunable magnetoelectric nanodevices

The perspective of energy-efficient and tunable functional magnetic nanostructures has triggered research efforts in the fields of voltage control of magnetism and spintronics. We investigate the magnetotransport properties of nanocomposite iron oxide/iron thin films with a nominal iron thickness of 5-50 nm and find a positive magnetoresistance at small thicknesses. The highest magnetoresistance was found for 30 nm Fe with +1.1% at 3 T. This anomalous behavior is attributed to the presence of Fe3O4-Fe nanocomposite regions due to grain boundary oxidation. At the Fe3O4/Fe interfaces, spin-polarized electrons in the magnetite can be scattered and reoriented. A crossover to negative magnetoresistance (−0.11%) is achieved at a larger thickness (>40 nm) when interface scattering effects become negligible as more current flows through the iron layer. Electrolytic gating of this system induces voltage-triggered redox reactions in the Fe3O4 regions and thereby enables voltage-tuning of the magnetoresistance with the locally oxidized regions as the active tuning elements. In the low-magnetic-field region (<1 T), a crossover from positive to negative magnetoresistance is achieved by a voltage change of only 1.72 V. At 3 T, a relative change of magnetoresistance about −45% during reduction was achieved for the 30 nm Fe sample. The present low-voltage approach signifies a step forward to practical and tunable room-temperature magnetoresistance-based nanodevices, which can boost the development of nanoscale and energy-efficient magnetic field sensors with high sensitivity, magnetic memories, and magnetoelectric devices in general