On a Structure Defined by a Tensor Field F of Type (1, 1) Satisfying F2 = 0

1973 ◽  
Vol 16 (3) ◽  
pp. 447-449
Author(s):  
C. S. Houh
Keyword(s):  
Tangent Space ◽  
Tensor Field ◽  
Class C ◽  
Almost Tangent Structure

Professor Eliopoulous studied almost tangent structure on manifolds M2n in [1], [2], [3]. An almost tangent structure F is a field of class C∞ of linear operations on M2n such that at each point x in M2n, Fx maps the complexified tangent space into itself and that Fx is of rank n everywhere and satisfies F2=0.

On a Riemannian Manifold M 2n with an Almost Tangent Structure

1969 ◽  
Vol 12 (6) ◽  
pp. 759-769 ◽  
Author(s):  
C.S. Houh
Keyword(s):  
Riemannian Manifold ◽  
Tangent Space ◽  
Tensor Field ◽  
Class C ◽  
Almost Tangent Structure

Professor Eliopoulous studied almost tangent structures on manifolds M2n in [1; 2]. An almost tangent structure F is a field of class C∞ of linear operations on M2n such that at each point x in M2n, Fx maps the complexified tangent space into itself and that Fx is of rank n everywhere and satisfies that F2 = 0. In this note, we consider a (1,1) tensor field . on a Riemannian M2n which satisfies everywhere and is such that the rank of F is n everywhere. Such gives an almost tangent structure F on M2n.

scholarly journals A characterization of locally homogeneous Riemann manifolds of dimension 3

1991 ◽  
Vol 123 ◽  
pp. 77-90 ◽  
Author(s):  
Kazuo Yamato
Keyword(s):  
Tensor Field ◽  
Local Homogeneity ◽  
Covariant Derivatives ◽  
Riemann Manifolds ◽  
Locally Homogeneous ◽  
Linear Isometries

It is classical to characterize locally homogeneous Riemann manifolds by infinitesimal conditions. For example, [Si] asserts that the local-homogeneity is equivalent to the existence of linear isometries between tangent spaces which preserve the curvatures and their covariant derivatives up to certain orders. It is also known that the local homogeneity is equivalent to the existence of a certain tensor field of type (1, 2) (for this and a further study, see [TV]).

scholarly journals PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

2021 ◽  
pp. 157
Author(s):  
Mohammad Nazrul Islam Khan ◽  
Lovejoy S. Das
Keyword(s):  
Tangent Space ◽  
Tensor Field ◽  
Vertical Structure ◽  
Product Structure ◽  
Lagrangian Manifold ◽  
Almost Product Structure ◽  
Vertical Tangent ◽  
B2 Structure

This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying (Fv2-a2)(Fv2+a2)(Fv2 - b2)(Fv2 + b2) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2n-dimensional Lagrange manifold E is defined and the F(±a2; ±b2)-structure on the vertical tangent space TV (E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on T(E). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r-parallel, r¯ anti half parallel when r = r¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

scholarly journals A theorem on polynomial lorentz structures

1986 ◽  
Vol 28 (2) ◽  
pp. 229-235
Author(s):  
Krzysztof Deszyński
Keyword(s):  
Tensor Field ◽  
Vector Fields ◽  
Minimal Polynomial ◽  
Differentiable Manifold ◽  
Real Numbers ◽  
Lorentz Structure ◽  
Polynomial Structure ◽  
Image Position

Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).We shall suppose that for any x ∈ Mis the minimal polynomial of the endomorphism fx: TxM → TxM.We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signaturesuch that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

Selfadjoint Metrics on Almost Tangent Manifolds Whose Riemannian Connection is Almost Tangent

1975 ◽  
Vol 17 (5) ◽  
pp. 671-674
Author(s):  
D. S. Goel
Keyword(s):  
Tensor Field ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Riemannian Metric ◽  
Almost Complex Structure ◽  
Constant Rank ◽  
Real Constant ◽  
Almost Complex ◽  
Riemannian Connection ◽  
Almost Tangent Structure

Let M be a differentiable manifold of class C∞, with a given (1, 1) tensor field J of constant rank such that J2=λI (for some real constant λ). J defines a class of conjugate (G-structures on M. For λ>0, one particular representative structure is an almost product structure. Almost complex structure arises when λ<0. If the rank of J is maximum and λ=0, then we obtain an almost tangent structure. In the last two cases the dimension of the manifold is necessarily even. A Riemannian metric S on M is said to be related if one of the conjugate structures defined by S has a common subordinate structure with the G-structure defined by S. It is said to be J-metric if the orthogonal structure defined by S has a common subordinate structure.

Crystalloid Inclusions in Hepatocyte Mitochondria and Their Similarity to Cytoplasmic Crystalloids

1974 ◽  
Vol 32 ◽  
pp. 198-199
Author(s):  
Odell T. Minick ◽  
Hidejiro Yokoo
Keyword(s):  
Liver Disease ◽  
Alcoholic Liver Disease ◽  
Special Interest ◽  
Structural Variations ◽  
Mitochondrial Alterations ◽  
The Matrix ◽  
Crystalloid Inclusions ◽  
Liver Biopsies

Mitochondrial alterations were studied in 25 liver biopsies from patients with alcoholic liver disease. Of special interest were the morphologic resemblance of certain fine structural variations in mitochondria and crystalloid inclusions. Four types of alterations within mitochondria were found that seemed to relate to cytoplasmic crystalloids.Type 1 alteration consisted of localized groups of cristae, usually oriented in the long direction of the organelle (Fig. 1A). In this plane they appeared serrated at the periphery with blind endings in the matrix. Other sections revealed a system of equally-spaced diagonal lines lengthwise in the mitochondrion with cristae protruding from both ends (Fig. 1B). Profiles of this inclusion were not unlike tangential cuts of a crystalloid structure frequently seen in enlarged mitochondria described below.

Evidence for a shear mechanism for the precipitation of γ-zirconium hydride in zirconium

1978 ◽  
Vol 36 (1) ◽  
pp. 658-659
Author(s):  
G.J.C. Carpenter
Keyword(s):  
Elevated Temperature ◽  
Strain Field ◽  
Zirconium Hydride ◽  
Zirconium Atom ◽  
Atom Diffusion ◽  
Solvus Temperature ◽  
Hexagonal Close Packed ◽  
Partial Dislocations ◽  
The Face

In zirconium-hydrogen alloys, rapid cooling from an elevated temperature causes precipitation of the face-centred tetragonal (fct) phase, γZrH, in the form of needles, parallel to the close-packed <1120>zr directions (1). With low hydrogen concentrations, the hydride solvus is sufficiently low that zirconium atom diffusion cannot occur. For example, with 6 μg/g hydrogen, the solvus temperature is approximately 370 K (2), at which only the hydrogen diffuses readily. Shears are therefore necessary to produce the crystallographic transformation from hexagonal close-packed (hep) zirconium to fct hydride.The simplest mechanism for the transformation is the passage of Shockley partial dislocations having Burgers vectors (b) of the type 1/3<0110> on every second (0001)Zr plane. If the partial dislocations are in the form of loops with the same b, the crosssection of a hydride precipitate will be as shown in fig.1. A consequence of this type of transformation is that a cumulative shear, S, is produced that leads to a strain field in the surrounding zirconium matrix, as illustrated in fig.2a.

Direct structure images of 30° partial dislocation cores in silicon

1987 ◽  
Vol 45 ◽  
pp. 242-243
Author(s):  
J. C. Barry ◽  
H. Alexander
Keyword(s):  
Dislocation Core ◽  
Preparation Technique ◽  
Burgers Vector ◽  
Vector Type ◽  
Sample Preparation Technique ◽  
Lattice Images ◽  
Dislocation Core Structure ◽  
Type Lattice ◽  
Direct Inspection

Dislocations in silicon produced by plastic deformation are generally dissociated into partials. 60° dislocations (Burgers vector type 1/2[101]) are dissociated into 30°(Burgers vector type 1/6[211]) and 90°(Burgers vector type 1/6[112]) dislocations. The 30° partials may be either of “glide” or “shuffle” type. Lattice images of the 30° dislocation have been obtained with a JEM 100B, and with a JEM 200Cx. In the aforementioned experiments a reasonable but imperfect match was obtained with calculated images for the “glide” model. In the present experiment direct structure images of 30° dislocation cores have been obtained with a JEOL 4000EX. It is possible to deduce the 30° dislocation core structure by direct inspection of the images. Dislocations were produced by compression of single crystal Si (sample preparation technique described in Alexander et al.).

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