Collineations of the curvature tensor in general relativity

Curvature collineations for the curvature tensor, constructed from a fundamental Bianchi Type-V metric, are studied. We are concerned with a symmetry property of space-time which is called curvature collineation, and we briefly discuss the physical and kinematical properties of the models.     

Mappings of empty space-times leaving the curvature tensor invariant

It is shown that if in some local coordinate system the componentsR ⁱ _jkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorR ⁱ _jkl invariant are found for each class of these space-times.     

Classification of the space-time curvature tensor,restricted case

Einstein field equations will be assumed in a certain restricted form. The problem to be solved consists, roughly, of the enumeration of all curvature tensors which satisfy these equations. Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Tripartite entanglement transformations and tensor rank

A basic question regarding quantum entangled states is whether one can be probabilistically converted to another through local operations and classical communication exclusively. While the answer for bipartite systems is known, we show that for tripartite systems, this question encodes some of the most challenging open problems in mathematics and computer science. In particular, we show that there is no easy general criterion to determine the feasibility, and in fact, the problem is NP hard. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the maximum rate to convert the Greenberger-Horne-Zeilinger state to a triangular distribution of three EPR states. Our results are based on connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory.     

Some potentials for the curvature tensor on three-dimensional manifolds

We study equations of Riemann–Lanczos type on three dimensional manifolds. Obstructions to global existence for global Lanczos potentials are pointed out. We check that the imposition of the original Lanczos symmetries on the potential leads to equations which do not have a determined type, leading to problems when trying to prove global existence. We show that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions. A second order potential for the Ricci tensor is introduced.     

Decomposition of the curvature tensor of hyper-Kähler manifolds

We give the decomposition into irreducible components of the curvature tensor of a hyper-Kähler manifold.     

Magnetic field correlation tensor in spaces of constant curvature

The concept of statistical homogeneity and isotropy for vector fields in spatial sections of constant curvature was analyzed. Solenoidality conditions for a corresponding correlation tensor were obtained for positive and negative curvature. It was shown that these conditions differ from the corresponding condition for fields in Euclidean space.     

Darboux transformations on timelike constant mean curvature surfaces

We give loop group theoretic reformulated Bäcklund transformations on constant mean curvature timelike surfaces in Minkowski 3-space. Further we present 1-soliton surfaces explicitly.     

Riemann curvature tensor in nonholonomic coordinates and non-Riemannian space-times

The calculation of the Riemann Curvature from the deviation of a vector undergoing parallel transport around a closed loop takes a very simple form when expressed in generalized geometrical notation. We discuss the parallel transport of a vector and the type of closed loop used in the calculation. Our method generalizes similar work of Morganstern, to nonholonomic coordinates and non-Riemannian space-times.     

Weyl,curvature, ricci,and metric tensor symmetries

Weyl symmetries for some specific spherically symmetric static spacetimes are derived and compared with metric, Ricci, and curvature tensor symmetries.     

Quantization on a two-dimensional phase space with a constant curvature tensor

Some properties of the star product of the Weyl type (i.e., associated with the Weyl ordering) are proved. Fedosov construction of the *-product on a two-dimensional phase space with a constant curvature tensor is presented. Eigenvalue equations for momentum p and position q on a two-dimensional phase space with constant curvature tensors are solved.     

Complex conformal rescalings and complex Lorentz transformations

Complex Lorentz transformations and complex conformal rescalings with independent conformal factors and are investigated in terms of elements of the group GL(2,C) G (2,C). It is shown how a general element of this group decomposes into a standard conformal rescaling (with =), a pure spin transformation, complex null rotations, and a complex boost-rotation. Of particular interest are the pure spin transformations that leave invariant the metric but transform the permutation spinors. It is these transformations that, when , are responsible for seemingly complicating the transformation law of the derivative operator and of spinors dependent thereon. It has been suggested that to avoid this complication one should allow the rescaled metric to have torsion. It is argued here that simplicity can be achieved even when the torsion-free condition is imposed.     

Complex quantum enveloping algebras as twisted tensor products

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universalR-matrices and give its relation to subgroups in the dual Hopf algebra.     

A BRST-like operator for space with zero curvature but non-zero torsion tensor

A new type of BRST-like operatorQ has been constructed for spaces with zero Riemann curvature tensor but with non-zero torsion. It is invariant under local coordinate transformation, and its proof depends upon the validity of the first Bianchi identity. Also, there exists a supersymmetry associated withQ. Finally, the cohomology induced byQ has been investigated.     

Analytical expression for the complex radius of curvature tensor Q for generalized gaussian beams

An expression for the complex radius of curvature tensor describing every kind of gaussian beam is proposed. This expression can be used in the tensor ABCD law. The circular, orthogonal astigmatic and non orthogonal astigmatic beams are expressed by means of this tensor.     

Complex ABCD transformations for optical beams with complex astigmatism

The method of the ray 4 × 4 ABCD matrices is extended to the three-dimensional optical systems of first order possessing complex astigmatism and containing selective elements with loss and field gain. These systems are found to have complex matrix elements. Properties of such matrices are studied, and a number of examples are considered that are of practical interest. It is shown that in the systems under consideration, like in the systems without loss and gain, such matrices retain the symplecticity property, on which the use of the method applied to specific models of optical-wave propagation is based.