Geometry of the submanifoldsESX n . III. Parallelism inESX n and its submanifolds

A connection which is both Einstein and semisymmetric is called anES connection. And a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through anES connection, is called ann-dimensionalES manifold and denoted byESX _n . This paper is the third part of a systematic study of the submanifoldsX _m ofESX _n . In the first part, we introduced a new concept of theC-nonholonomic frame of reference inESX _n at points ofX _m and dealt with its consequences. In the second part, the generalized fundamental equations on a hypersubmanifold ofESX _n were derived as an application of theC-nonholonomic frame of reference. The purpose of the present paper is to study parallelism inESX _n and in its submanifoldX _m , using theC-nonholonomic frame of reference and the new concept ofES _i curves.     

Geometry of the submanifolds of SEXn. II. The generalized fundamental equations for the hypersubmanifold of SEXn

A connection which is both Einstein and semisymmetric is called an SE connection, and a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through an SE connection is called ann-dimensional SE manifold and denoted by SEX_n. This paper is a direct continuation of earlier work. In this paper, we derive the generalized fundamental equations for the hypersubmanifold of SEX_n, including generalized Gauss formulas, generalized Weingarten equations, and generalized Gauss-Codazzi equations.     

Geometry of the submanifolds of SEXn. I. The C-nonholonomic frame of reference

A connection which is both Einstein and semisymmetric is called an SE connection. A generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through an SE connection is called ann-dimensional SE manifold and denoted by SEX_n. This paper is the introductory part of a systematic study of the submanifolds of SEX_n. It introduces a new concept of the C-nonholonomic frame of references in SEX_n at points of its submanifold and deals with its consequences. The second part will deal with the generalized fundamental equations on an SE hypersubmanifold of SEX_n. The third part will be devoted to the theory of parallelism in SEX_n and in its submanifold. Finally, the last part will study the curvature theory in a submanifold of SEX_n.     

Curvature tensors unified field equations on SEXn

We study the curvature tensors and field equations in then-dimensional SE manifold SEX_n. We obtain several basic properties of the vectorsS andU and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field equations is discussed in SEX_n and one of its particular solutions is constructed and displayed.     

Field equations onSE(k)-manifoldX n

An Einstein connection which is both a special connection and a (k)-connection is called anSE(k)-connection. And a generalized even-dimensional Riemannian manifoldX _n with the so-called SE(k)-condition defined by theSE(k)-connection is called theSE(k)-manifold. We obtain the necessary and sufficient condition that there is a uniqueSE(k)-connection inX _n . Next, using these results, we define theSE(k)-manifold and study the properties of the curvature tensors and the field equations in theSE(k)-manifoldX _n .     

Precision PAC measurements in Er2O3 and Ho2O3 single crystals and structure refinement

The structure of C-form Ho₂O₃ and Er₂O₃ single crystals and powder samples was investigated by the electric quadrupole hyperfine interaction of¹¹¹In(EC)¹¹¹Cd probe ions using the perturbed - angular correlation method (PAC). The resulting set of refined atomic coordinates is compared to X-ray data and used to calculate the orientations of the electric field gradients (EFG) which are reproduced by the PAC measurements in single crystals. The temperature dependence of the coordinates was measured for both substances.     

Relativistic geometrical optics

We investigate the gravitational and electromagnetic fields on the generalized Lagrange space endowed with the metricg _ij(x, y) = _ij(x) + {1 + 1/n ² (x, y)}y _iy_j. The generalized Lagrange spacesM ^m do not reduce to Lagrange spaces. Consequently, they cannot be studied by methods of symplectic geometry. The restriction of the spacesM ^m to a sectionS (M) leads to the Maxwell equations and Einstein equations for the electromagnetic and gravitational fields in dispersive media with the refractive indexn(x, V) endowed with the Synge metric. Whenn(x, V) = 1 we have the classical Einstein equations. If 1/n ²=1–1/c ² (c being the light velocity), we get results given previously by the authors. The present paper is a detailed version of a work in preparation.     

On the variation of hopping transport in amorphous silicon with preparation conditions

Thedc conductivity of amorphous silicon prepared by two successive ion bombardments at different temperatures has been measured as a function of temperature. The results may be expressed in terms of a generalized hopping formula =₀ exp [–(T ₀/T) ⁿ where the parameter set {n,T ₀, ₀} varies with the irradiation conditions. In particular, the hopping exponent has been found to assume the limiting values ofn1/4 at irradiation temperatures ofT _i100 K and ofn1/2 atT _i500 K, whereas intermediate values ofn have been observed for temperatures inbetween. It is concluded that thermally activated redistribution processes of radiation defects control the final state of disorder in the irradiated samples, which in turn determines the particular hopping characteristics. Within the framework of existing theories the two limiting cases can be explained to be due to a disordered solid of homogeneous and granular structure, respectively.     

Quantum Even Spheres Σ<Superscript><Emphasis Type="Italic">2n</Emphasis></Superscript><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript> from Poisson Double Suspension

We define even dimensional quantum spheres ²ⁿ _q that generalize to higher dimension the standard quantum two-sphere of Podle and the four-sphere ⁴ _q obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podle two-sphere. The Poisson spheres that we get have the same kind of symplectic foliation consisting of a degenerate point and a symplectic ²ⁿ and, after quantization, have the same C ^* –algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.     

Densité des vecteurs propres généralisés d'une classe d'opérateurs compacts non auto-adjoints et applications

We consider a closed densely defined linear operatorT in a Hilbert spaceE, and assume the existence of ₀ (T) such thatK = (T - ₀ I)^-1 is compact and the existence ofp>0 such thats _n (K)=o((n ^–1/p)), whereS _n (K) denotes the sequence of non-zero eigenvalues of the compact hermitian operator . In this work, sufficient conditions (announced in [1]) are introduced to assure that the closed subspace ofE spanned by the generalized eigenvectors ofT coincides withE. These conditions are in particular verified by a family of non-self-adjoint operators arising in reggeon's field theory.     

Critical current density of a YBa2Cu3O7 film: Comparison between experiment and collective pinning theory

With the vibrating reed technique we have measured the critical current densityJ _c of a Y 123 200 nm thin film as a function of magnetic field perpendicular to the CuO₂ planes (O T<B8 T) and temperature (20KT60 K). At fieldsB<0.1 TJ _c is magnetic field independent and decreases at higher fields. A comparison with theory indicates that a crossover from a single pinning to a small bundle collective pinning regime may explain the observed behaviour. According to our estimate the main pinning centers are weak point pins due to oxygen vacancies. From the temperature dependence ofJ _c atB0 we obtain a temperature dependence of the thermodynamical critical fieldB _c (1–T/T _c )² forT20 K which agrees with the anomalous temperature dependence ofB _c2 observed recently in highly anisotropic high temperature superconductors.     

Einstein and Brans–Dicke Frames in Multidimensional Cosmology

Inhomogeneous multidimensional cosmological models with a higher-dimensional space-time manifold _{0 i=1} ⁿ M_i (n 1) are in stigated under dimensional reduction to a D ₀-dimensional effective non-minimally coupled -model which generalizes the familiar Brans–Dicke model. The general form of the Einstein frame representation of multidimensional solutions known in the Brans–Dicke frame is given with respect to cosmic synchronous time. As an example, the transformation is demonstrated explicitly for the generalized Kasner solutions where it is shown that solutions in the Einstein frame show no inflation of the external space although they can undergo deflation after the cosmic synchronous time inversion.     

Space-time as a micromorphic continuum

We review generally-covariant Lagrangians for the field of linear coframes in ann-dimensional manifold. Discussed are Lagrangians invariant under the internal groupGL(n, ) and under its pseudo-Euclidean subgroups. It is shown that group spaces of semisimple Lie groups and certain of their modifications are natural vacuumlike solutions for allGL(n, )-invariant models. In some sense the signature of space-time may be interpreted as a consequence of differential equations; the velocity of light is an integration constant.     

Neighbours of Einstein's equations: Connections and curvatures

Once the action for Einstein's equations is rewritten as a functional of anSO(3, ) connection and a conformal factor of the metric, it admits a family of neighbours having the same number of degrees of freedom and a precisely defined metric tensor. This paper analyzes the relation between the Riemann tensor of that metric and the curvature tensor of theSO(3) connection. The relation is very complicated in general. The Einstein case is distinguished by the fact that two naturalSO(3) metrics on theGL(3) fibres coincide. In the general case the theory is bimetric on the fibres.     

Affine connection structure of the charged symplectic 2-form

It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion-free and the antisymmetric part of theR ^4* translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravitational and electromagnetic fields can therefore be reformulated as aP(4)=O(1, 3)R ^4* geometric theory with phase space the affine cotangent bundleAT ^* M of spacetime. The sourcefree Maxwell equations are reformulated as a pair of geometrical conditions on the ^4* curvature that are exactly analogous to the source-free Einstein equations.     

Quotient space solutions of 11-dimensional supergravity

All the Freund-Rubin type solutions of the 11-dimensional supergravity with a simply connected quotient spaceG/H as the compact 7-dimensional manifold are found. Their geometries depend only on the imbedding ofHG and the Riemannian structure ofG/H. In particular, SU₃×SU₂×U₁/SU₂×U₁×U₁, Einstein solutions are discussed in detail.     

Three- and five-dimensional considerations of the geometry of Einstein's* g-unified field theory

We study the algebraic and differential geometric structures of three- and five-dimensional^* g-unified field theory, with emphasis on the five-dimensional^* g-unified field theory, in which we derive a new set of powerful recurrence relations which hold in a five-dimensional generalized Riemannian manifoldX ₅ , prove a necessary and sufficient condition for the existence and uniqueness of the solution of the system of the Einstein equations in the first two classes, and find a precise tensorial representation of the Einstein connection in terms of^*g .     

Harmonic-free and fixed-exit monolithic X-ray monochromator

It is shown that if the second wall of a channel-cut crystal x-ray monochromator is properly curved, the monochromator is harmonic free for all. Moreover, if the first wall is suitably inclined to the diffracting planes, the position of the exit beam may be kept constant for all. This requires a correction of the position of the monochromator for each by its axial and radial translation.     

A Canonical Null Gauge for the Einstein Equations

Associated with any choice of outgoing null (characteristic) coordinate, we construct a null tetrad which is determined uniquely by purely local information. Unlike other well-known characteristic gauge conditions [1–4], this canonical tetrad is determined at any point solely by the metric and its first derivatives at that point. The tetrad leads to a radial coordinate which is also uniquely determined by purely local information. These properties greatly simplify the structure of the hypersurface Einstein equations and may be used to directly compare metrics with compatible null foliations.