An introductory example: the uniform static field

This chapter discusses some physical effects related to two simple metrics: the RIndler metric and the uniform static field. The purpose is to illustrate the methods by applying them in an exact calculation which is not too taxing. The Christoffel symbols and curvature tensors are obtained, and some example geodesics are calculated. The force experienced by a fisherman fishing in the RIndler metric is calculated.

LOCALLY ORTHOTROPIC LAY-UP AS AN OPTIMAL SOLUTION FOR VAT POST-BUCKLED COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE VON KARMAN LIMITS

The present paper deals with the optimization of post-buckled VAT (variable angle tow) composite plates with large deflections. The Kirchhoff assumptions are used. The plates have a symmetric lay-up. The large deflection geometrically nonlinear theory above the von Karman limits is employed. The structural potential energy is treated as a measure of structural stiffness. For the plate stiffness maximization problem, the first-order necessary conditions of the local optimality are derived. The mathematical analysis of the conditions is performed. The conditions contain two terms. One of them corresponds to the mid-plane strains; another one corresponds to the generalized plate curvatures. A locally orthotropic lay-up is identified as an optimal solution. The local ply material direction is clearly coupled with the principal directions of 2D-strains and generalized curvatures. A particular solution of the linear combination of the ply optimality conditions is indicated. For the solution two pairs of the structural tensors are co-axial: the force and the strain tensors, as well as the moment and the generalized curvature tensors.

$D_a$-HOMOTHETIC DEFORMATION AND RICCI SOLITIONS IN THREE DIMENSIONAL QUASI-SASAKIAN MANIFOLDS

In the present paper, we have studied curvature tensors of a quasi-Sasakian manifold with respect to the $D_a$-homothetic deformation. We have deduced the Ricci solition in quasi-Sasakian manifold with respect to the $D_a$-homothetic deformation. We have also proved that the quasi-Sasakian manifold is not $\bar{\xi}$-projectively flat under $D_a$-homothetic deformation. Also we give an example to prove the existance of quasi-Sasakian manifold.

A geometrical model for the evolution of spherical planetary nebulae based on thin-shell formalism

A spherical planetary nebula is described as a geometric model. The nebula itself is considered as a thin-shell, which is visualized as a boundary of two spacetimes. The inner and outer curvature tensors of the thin-shell are found in order to get an expression of the energy-momentum tensor on the thin-shell. The energy density and pressure expressions are derived using the energy-momentum tensor. The time evolution of the radius of the thin-shell is obtained in terms of the energy density. The model is tested by using a simple power function for decreasing energy density and the evolution pattern of the planetary nebula is attained.

On the α-connections and the α-conformal equivalence on statistical manifolds

PurposeIn this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^.Design/methodology/approachIn the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples.FindingsWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.Originality/valueWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

Coupling of quantum gravitational field with Riemann and Ricci curvature tensors

The theoretical problem of establishing the coupling properties existing between the classical and quantum gravitational field with the Ricci and Riemann curvature tensors of General Relativity is addressed. The mathematical framework is provided by synchronous Hamilton variational principles and the validity of classical and quantum canonical Hamiltonian structures for the gravitational field dynamics. It is shown that, for the classical variational theory, manifestly-covariant Hamiltonian functions expressed by either the Ricci or Riemann tensors are both admitted, which yield the correct form of Einstein field equations. On the other hand, the corresponding realization of manifestly-covariant quantum gravity theories is not equivalent. The requirement imposed is that the Hamiltonian potential should represent a positive-definite quadratic form when performing a quadratic expansion around the equilibrium solution. This condition in fact warrants the existence of positive eigenvalues of the quantum Hamiltonian in the harmonic-oscillator representation, to be related to the graviton mass. Accordingly, it is shown that in the background of the deSitter space-time, only the Ricci tensor coupling is physically admitted. In contrast, the coupling of quantum gravitational field with the Riemann tensor generally prevents the possibility of achieving a Hamiltonian potential appropriate for the implementation of the quantum harmonic-oscillator solution.