scholarly journals On the converse of Weyl’s conformal and projective theorems

2013 ◽  
Vol 94 (108) ◽  
pp. 55-65 ◽  
Author(s):  
Graham Hall
Keyword(s):  
Positive Definite ◽  
The Other ◽  
Dimensional Problem ◽  
Projective Tensor ◽  
Definite Case

This note investigates the possibility of converses of the Weyl theorems that two conformally related metrics on a manifold have the same Weyl conformal tensor and that two projectively related connections on a manifold have the same Weyl projective tensor. It shows that, in all relevant cases, counterexamples to each of Weyl?s theorems exist except for his conformal theorem in the 4-dimensional, positive definite case, where the converse actually holds. This (conformal) 4-dimensional problem is then solved completely for the other possible signatures.

scholarly journals Prolongations of G-Structures to Tangent Bundles

1968 ◽  
Vol 32 ◽  
pp. 67-108 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Definite ◽  
The Other ◽  
Tensor Fields ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
The One ◽  
Lie Subgroup ◽  
Tangent Bundles

The purpose of the present paper is to study the prolongations of G-structures on a manifold M to its tangent bundle T(M), G being a Lie subgroup of GL(n,R) with n = dim M. Recently, K. Yano and S. Kobayashi [9] studied the prolongations of tensor fields on M to T(M) and they proposed the following question: Is it possible to associate with each G-structure on M a naturally induced G-structure on T(M), where G′ is a certain subgroup of GL(2n,R)? In this paper we give an answer to this question and we shall show that the prolongations of some special tensor fields by Yano-Kobayashi — for instance, the prolongations of almost complex structures — are derived naturally by our prolongations of the classical G-structures. On the other hand, S. Sasaki [5] studied a prolongation of Riemannian metrics on M to a Riemannian metric on T(M), while the prolongation of a (positive definite) Riemannian metric due to Yano-Kobayashi is always pseudo-Riemannian on T(M) but never Riemannian. We shall clarify the circumstances for this difference and give the reason why the one is positive definite Riemannian and the other is not.

scholarly journals A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 213
Author(s):  
Herbert Dueñas Ruiz ◽  
Francisco Marcellán ◽  
Alejandro Molano
Keyword(s):  
Positive Definite ◽  
Linear Functionals ◽  
Definite Case ◽  
Coherent Pairs

In this paper, we study a classification of symmetric ( 1 , 1 ) -coherent pairs by using a symmetrization process. In particular, the positive-definite case is carefully described.

SUPERSYMMETRY IN 2+2 DIMENSIONS

2000 ◽  
Vol 15 (21) ◽  
pp. 1349-1355 ◽  
Author(s):  
F. T. BRANDT ◽  
D. G. C. MCKEON ◽  
T. N. SHERRY
Keyword(s):  
Hilbert Space ◽  
Positive Definite ◽  
The Other ◽  
Dirac Spinors

The analysis of spinors in 2+2 dimensions is reviewed. The two most basic supersymmetry algebras in this space are considered. In the simpler case, the supersymmetry generators are Majorana spinors. In the other algebra we consider, the supersymmetry generators are Dirac spinors. Consistency with having a positive definite Hilbert space is shown to be impossible in both of these particular cases.

scholarly journals Ideals of the Fourier algebra, supports and harmonic operators

2016 ◽  
Vol 161 (2) ◽  
pp. 223-235 ◽  
Author(s):  
M. ANOUSSIS ◽  
A. KATAVOLOS ◽  
I. G. TODOROV
Keyword(s):  
Short Proof ◽  
Positive Definite Function ◽  
Positive Definite ◽  
The Other ◽  
Definite Function ◽  
Fourier Algebra ◽  
Schur Multipliers ◽  
The Common

AbstractWe examine the common null spaces of families of Herz–Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in [1] can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.

scholarly journals DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER -SPHERES OF CONSTANT CURVATURE

2021 ◽  
pp. 1-32
Author(s):  
Christian Lange ◽  
Thomas Mettler
Keyword(s):  
Ricci Curvature ◽  
Constant Curvature ◽  
Projective Spaces ◽  
Positive Definite ◽  
The Other ◽  
Duality Result ◽  
Veronese Embedding ◽  
Other Hand ◽  
The One ◽  
Weyl Connections

Abstract We establish a one-to-one correspondence between, on the one hand, Finsler structures on the $2$ -sphere with constant curvature $1$ and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb {CP}(a_1,a_2)\rightarrow \mathbb {CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$ -spheres of constant curvature whose geodesics are all closed.

scholarly journals VON NEUMANN ENTROPY AND RELATIVE POSITION BETWEEN SUBALGEBRAS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350066 ◽  
Author(s):  
MARIE CHODA
Keyword(s):  
Tensor Product ◽  
Relative Position ◽  
Von Neumann Algebra ◽  
Positive Definite ◽  
The Other ◽  
Von Neumann Entropy ◽  
Product Decomposition ◽  
Positive Definite Matrices ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

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