The spinor and tensor fields with higher spin on spaces of constant curvature

In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin $$j+1/2$$ j + 1 / 2 over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power $$j+1$$ j + 1 and understand how the spinor fields with spin $$j+1/2$$ j + 1 / 2 are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature.

Dirac particle in the external coulomb field on the background of the Lobachevsky–Riemann space models

The known systems of the radial equations describing the hydrogen atom on the basis of the Dirac equation in the Lobachevsky–Riemann spaces of constant curvature are investigated. In the both geometrical models, the differential equations of second order with six regular singular points are found, and their exact solutions of Frobenius type are constructed. To produce the quantization rule for energy values we use the known condition which separates the transcendental Frobenius solutions. This provides us with the energy spectra that are physically interpretable and are similar to those for the Klein–Fock–Gordon particle in these space models. These spectra are similar to those that previously have appeared in studying the same systems of the equations with the use of the semi-classical approximation.

Special semi-reducible pseudo-Riemannian spaces

The paper contains necessary conditions allowing to reduce matrix tensors of pseudo-Riemannian spaces to special forms called semi-reducible, under assumption that the tensor defining tensor characteristic of semireducibility spaces, is idempotent. The tensor characteristic is reduced to the spaces of constant curvature, Ricci-symmetric spaces and conformally flat pseudo-Riemannian spaces. The obtained results can be applied for construction of examples of spaces belonging to special types of pseudo-Riemannian spaces. The research is carried out locally in tensor shape, without limitations imposed on a sign of a metric.

Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold

PurposeThe author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.Design/methodology/approachThe author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.FindingsThe author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).Originality/valueTo the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

Volumes of Polyhedra in Non-Euclidean Spaces of Constant Curvature

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute).In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in 3-dimensional spherical and hyperbolic spaces of constant curvature K = 1 and K = -1, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincare model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

Universal p-form black holes in generalized theories of gravity

We explore how far one can go in constructing d-dimensional static black holes coupled to p-form and scalar fields before actually specifying the gravity and electrodynamics theory one wants to solve. At the same time, we study to what extent one can enlarge the space of black hole solutions by allowing for horizon geometries more general than spaces of constant curvature. We prove that a generalized Schwarzschild-like ansatz with an arbitrary isotropy-irreducible homogeneous base space (IHS) provides an answer to both questions, up to naturally adapting the gauge fields to the spacetime geometry. In particular, an IHS–Kähler base space enables one to construct magnetic and dyonic 2-form solutions in a large class of theories, including non-minimally couplings. We exemplify our results by constructing simple solutions to particular theories such as $$R^2$$ R 2 , Gauss–Bonnet and (a sector of) Einstein–Horndeski gravity coupled to certain p-form and conformally invariant electrodynamics.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.