Tetraplectic structures, tri-momentum maps, and quaternionic flag manifolds

The purpose of this note is to define tri-momentum maps for certain manifolds with an Sp(1)ⁿ-action. We exhibit many interesting examples of such spaces using quaternions. We show how these maps can be used to reduce such manifolds to ones with fewer symmetries. The images of such maps for quaternionic flag manifolds, which are defined using the Dieudonné determinant, resemble the polytopes from the complex case.     

Generalized Einstein manifolds

A Finslerian manifold is called a generalized Einstein manifold (GEM) if the Ricci directional curvature R(u,u) is independent of the direction. Let F⁰(M, g_t) be a deformation of a compact n-dimensional Finslerian manifold preserving the volume of the unitary fibre bundle W(M). We prove that the critical points g₀ F⁰(g_t) of the integral I(g_t) on W(M) of the Finslerian scalar curvature (and certain functions of the scalar curvature) define a GEM. We give an estimate of the eigenvalues of Laplacian Δ defined on W(M) operating on the functions coming from the base when (M, g) is of minima fibration with a constant scalar curvature H admitting a conformal infinitesimal deformation (CID). We obtain λ ≥ H/(n − 1) (Δf = λf). If M is simply connected and λ = H/(n − 1), then (M, g) is Riemannian and is isometric to an n-sphere. We first calculate, in the general case, the formula of the second variationals of the integral I (g_t) for G = g₀, then for a CID we show that for certain Finslerian manifolds, I″(g₀) > 0. Applications to the gravitation and electromagnetism in general relativity are given. We prove that the spaces characterizing Einstein-Maxwell equations are GEMs.     

Constant scalar curvature and warped product globally null manifolds

This paper deals with the curvature properties of a class of globally null manifolds (M,g) which admit a global null vector field and a complete Riemannian hypersurface. Using the warped product technique we study the fundamental problem of finding a warped function such that the degenerate metric g admits a constant scalar curvature on M. Our work has an interplay with the static vacuum solutions of the Einstein equations of general relativity.     

Algebraic integrability conditions for Killing tensors on constant sectional curvature manifolds

We use an isomorphism between the space of valence two Killing tensors on an n$n$-dimensional constant sectional curvature manifold and the irreducible GL(n+1)$GL(n+1)$-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor, resulting in two simple algebraic equations of degree two and three. As a first application of this we construct a new family of integrable Killing tensors.     

Jacobi fields on statistical manifolds of negative curvature

Two entropic dynamical models are considered. The geometric structure of the statistical manifolds underlying these models is studied. It is found that in both cases, the resulting metric manifolds are negatively curved. Moreover, the geodesics on each manifold are described by hyperbolic trajectories. A detailed analysis based on the Jacobi equation for geodesic spread is used to show that the hyperbolicity of the manifolds leads to chaotic exponential instability. A comparison between the two models leads to a relation among statistical curvature, stability of geodesics and relative entropy-like quantities. Finally, the Jacobi vector field intensity and the entropy-like quantity are suggested as possible indicators of chaoticity in the ED models due to their similarity to the conventional chaos indicators based on the Riemannian geometric approach and the Zurek-Paz criterion of linear entropy growth, respectively.     

Lorentz manifolds modelled on a Lorentz symmetric space

We give examples of Lorentz manifolds modelled on an indecomposable Lorentz symmetric space which are geodesically complete and not locally homogeneous.     

On a scale function for testing the conformality of a Finsler manifold to a Berwald manifold

In this paper, we are going to discuss the problem whether how we can check the conformality of a Finsler manifold to a Berwald manifold. The method is based on a differential 1-form constructing on the underlying manifold by the help of integral formulas such that its exterior derivative is conformally invariant. If the Finsler manifold is conformal to a Berwald manifold, then the exterior derivative vanishes. This gives the following necessary condition: the differential form is closed and, at least locally, it is exact as the exterior derivative of a scale function for testing the conformality. A necessary and sufficient condition is also given in terms of a distinguished linear connection on the underlying manifold – it is expressed by the help of canonical data. In order to illustrate how we can simplify the process in special cases Randers manifolds are considered with some explicit calculations.     

Riemann curvature scalar of spacetime tangent bundle

The maximum possible proper acceleration relative to the vacuum determines much of the differential geometric structure of the space-time tangent bundle. By working in an anholonomic basis adapted to the spacetime affine connection, one derives a useful expression for the Riemann curvature scalar of the bundle manifold. The explicit documentation of the proof is important because of the central role of the curvature scalar in the formulation of an action with resulting field equations and associated solutions to physical problems.     

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka–Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only if it is flat.     

On a poisson reduction for Gel'fand-Zakharevich manifolds

We formulate and discuss a reduction theorem for Poisson pencils associated with a class of integrable systems, defined on bi-Hamiltonian manifolds, recently studied by Gel'fand and Zakharevich. The reduction procedure is suggested by the bi-Hamiltonian approach to the separation of variables problem.     

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.     

Volumes of compact manifolds

We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius condition for spheres. For this aim we first describe the manifold with some parameters, set up a metric, which induces a volume element, and perform the integration for the adequate range of the parameters; in most cases our manifolds will be either spheres or (twisted) products of spheres, or quotients of spheres (homogeneous spaces).Our results should be useful in several physical instances, as instanton calculations, propagators in curved spaces, sigma models, geometric scattering in homogeneous manifolds, density matrices for entangled states, etc. Some flag manifolds have also appeared recently as exceptional holonomy manifolds; the volumes of compact Einstein manifolds appear in string theory.     

On Randers spaces of scalar curvature

Randers spaces of scalar curvature or of constant curvature have not yet been studied completely because of the complicated forms of the connection coefficients and the curvature tensors. A necessary and sufficient condition for a Randers space to be of scalar curvature, found under some assumptions by this time, is given in this paper. Also a condition for the space to be of constant curvature, found previously in a complicated form, is given in simple form with a geometrical meaning. It is shown that a Randers space is locally Minkowskian if and only if it is a space of vanishing constant curvature.     

On curvature homogeneous three-dimensional Lorentzian manifolds

The aim of this paper is the study of three-dimensional Lorentzian manifolds whose Ricci tensor has three equal constant eigenvalues, whose associated eigenspace is two-dimensional. A complete local classification of this class of curvature homogeneous manifolds is presented. It turns out that, if the eigenvalue is zero, these are exactly the curvature homogeneous manifolds modelled on an indecomposable, non-irreducible Lorentzian symmetric space, which were first studied in Cahen etaal. (1990), and the techniques presented in this paper can therefore be applied to obtain a complete (local) classification of these manifolds, and to construct a number of new examples of such manifolds.     

On nonlinear higher spin curvature

We present the first nonlinear term of the higher spin curvature which is covariant with respect to deformed gauge transformations that are linear in the field. We consider the case of spin 3 after presenting spin 2 as an example, and then construct the general spin s quadratic term of the de Wit–Freedman curvature.     

On spherically symmetric Finsler metrics of scalar curvature

Spherically symmetric Finsler metrics form a rich class of Finsler metrics. In this paper we find equations that characterize spherically symmetric Finsler metrics of scalar flag curvature. By using these equations, we construct infinitely many non-projectively flat spherically symmetric Finsler metrics of scalar curvature.