Almost-product Structure Research Articles

AN ALMOST-PRODUCT STRUCTURE ON THE FOLIATED MANIFOLD R.

Adv. Math. Sci. J. Vol. 9, No. 12 Open Access AN ALMOST-PRODUCT STRUCTURE ON THE FOLIATED MANIFOLD R. S. Herinantenaina and H. S. G. Ravelonirina DOI: https://doi.org/10.37418/amsj.9.12.85 Full Text Abstract Abstract Let $M$ be a differentiable, connected, paracompact (resp. Compact) manifold of dimension $n+m$, of class $C^{infty}$; $L$ a vector valued $1-$form with … page AMSJ9-N12-85 Read More »

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds

We study two Einstein–Hilbert type actions on an almost-product metric-affine manifold, considered as functionals of the contorsion tensor. The first one is the total mixed scalar curvature of the linear connection, and the second one is based on a new type of curvature, recently introduced by B. Opozda for statistical structures. We deduce Euler–Lagrange equations of the actions and examine critical contorsion tensors associated with general and distinguished classes of connections, e.g. metric, statistical and adapted. The existence of such critical tensors depends on simple geometric properties of the almost-product structure, expressed only in terms of the Levi-Civita connection.

Two-dimensional supersymmetric sigma models on almost-product manifolds and non-geometry

In this paper we study the relation between the target space geometry of (1, 1) sigma models and the factorization of the associated superconformal theory, at the classical level. It turns out that the superconformal currents T± factorize as , with and conserved separately, provided that the target space admits projectors P(±) and Q(±) (that is, a pair of almost-product structures), compatible with the Riemannian structure of the target space, and covariantly constant with respect to the two torsionful connections ∇(±) that arise naturally from the sigma model. It is a surprising result that the integrability of the projectors is not an obstruction for the associated symmetries and to form copies of the superconformal algebra. While one expects to be able to define a superconformal theory associated with a particular Riemannian submanifold defined by an integrable projector, a consequence of the above result is that there are no obstructions to defining a superconformal theory associated with non-integrable projectors. We show that this notion of non-geometry encompasses the locally non-geometric examples that arise in the T-duality inspired doubled formulations. In addition, we derive the general conditions for (2, 2) supersymmetry to be realized in the projective sense. This extends the relation between (2, 2) sigma models and bi-Hermitian geometry to the non-geometric setting. For the bosonic subsector we propose a BRST-type approach to defining the physical degrees of freedom in the non-geometric scenario.

On the geometry of Killing and conformal tensors

The second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almost-product structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space–time cases 1+3 and 2+2 are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum Petrov-Bel type D solutions is offered. A generalization of these results to a wider family of type D space–times is also obtained.

A Rainich-Like Approach to the Killing-Yano Tensors

The Rainich problem for Killing-Yano tensors posed by Collinson [1] is solved. In intermediate steps, we first obtain the necessary and sufficient conditions for a 2+2 almost-product structure to determine the principal 2–planes of a Killing-Yano tensor. Then we give the additional conditions on a symmetric Killing tensor for it to be the square of a Killing-Yano tensor. We also analyze a similar problem for the conformal Killing-Yano tensors. Our results show that, in both cases, the principal 2–planes define a Maxwellian structure. The associated Maxwell fields are obtained and we outline how this approach is of interest in studying the spacetimes that admit these kind of first integrals of the geodesic equation.

The projector method for regular Poisson manifolds

Degenerate Lagrangians have recently been studied through a pair of projection operators induced by an almost-product structure []. We extend the projector method to regular Poisson manifolds. This approach proves to be useful in the interpretation of the theories of collective motion of N-particle systems for applications in nuclear physics.

A new look at degenerate Lagrangian dynamics from the viewpoint of almost-product structures

Singular Lagrangian systems are studied in the framework of almost-product structures. The choice of an appropriate almost-product structure permits us to obtain the dynamics. The relationship with the Dirac bracket is also elucidated.

A class of harmonic almost-product structures

The energy of Riemannian almost-product structure P is measured by forming the Dirichlet integral of the associated Gauss section γ, and P is decreed harmonic if γ criticalizes the energy functional when restricted to the submanifold of sections of the Grassman bundle. Euler-Lagrange equations are obtained, and geometrically transformed in the special case when P is totally geodesic. These are seen to generalize the Yang-Mills equations, and generalizations of the self-duality and anti-self-duality conditions are suggested. Several applications are then described. In particular, it is considered whether integrability of P is a necessary condition for γ to be harmonic.

Spaces of constant para-holomorphic sectional curvature

We consider the sectional curvatures for metric (J4 = 1)-manifolds, and study particularly the general expression of the metric and almostproduct structure in normal coordinates for para-Kaehlerian manifolds of constant para-holomorphic sectional curvature. We also introduce models of such spaces.

A new proof of the generalized Birkhoff theorem

A new proof of the generalized Birkhoff theorem in general relativity is presented. The partial results contained in previous proofs are recovered in a unified treatment of the different kinds of space-times to which the theorem applies. The proof is based on the fact that these space-times are conformal to the direct product of two two-dimensional manifolds (almost-product structure).

Characteristic classes of almost-flag structures

The aim of this paper is to present a basic theory of characteristic classes of almost-flag structures. Vanishing theorems for primary characteristic classes of almost-flag and almost-product structures are proved. Consequences of these theorems for secondary characteristic classes are drawn in the framework of Lehmann's theory. Two theorems on residues of flag structures complete the paper.

Riemannian almost-product structures with maximal mobility

The aim of the paper is to give a classification up to isomorphism of (local or simply-connected global) Riemannian almost-product structures (i.e. O(n1) × ... × O(nt)-structures) whose automorphism group has maximal dimension (the so-called ℳ-structures). The describing objects are Lie algebras with some additional structure, and the methods are mainly Lie algebraic. The results obtained are applied to determine all ℳ-structures on simply-connected spaces of constant curvature.