Isometry Research Articles

Carroll Symmetry, Dark Energy and Inflation

Carroll symmetry arises from Poincaré symmetry upon taking the limit of vanishing speed of light. We determine the constraints on the energy-momentum tensor implied by Carroll symmetry and show that for energy-momentum tensors of perfect fluid form, these imply an equation of state E+P=0 for energy density plus pressure. Therefore Carroll symmetry might be relevant for dark energy and inflation. In the Carroll limit, the Hubble radius goes to zero and outside it recessional velocities are naturally large compared to the speed of light. The de Sitter group of isometries, after the limit, becomes the conformal group in Euclidean flat space. We also study the Carroll limit of chaotic inflation, and show that the scalar field is naturally driven to have an equation of state with w = − 1. Finally we show that the freeze-out of scalar perturbations in the two point function at horizon crossing is a consequence of Carroll symmetry. To make the paper self-contained, we include a brief pedagogical review of Carroll symmetry, Carroll particles and Carroll field theories that contains some new material as well. In particular we show, using an expansion around speed of light going to zero, that for scalar and Maxwell type theories one can take two different Carroll limits at the level of the action. In the Maxwell case these correspond to the electric and magnetic limit. For point particles we show that there are two types of Carroll particles: those that cannot move in space and particles that cannot stand still.

Compactifications of horospheric products

We define and study a new compactification, called the height compactification of the horospheric product of two infinite trees. We will provide a complete description of this compactification. In particular, we show that this compactification is isomorphic to the Busemann compactification when all the vertices of both trees have degree at least three, which also leads to a precise description of the Busemann functions in terms of the points in the geometric compactification of each tree. We will discuss an application to the asymptotic behavior of integrable ergodic cocycles with values in the isometry group of such horospheric product.

Entire constant mean curvature graphs in ℍ2× ℝ

For $0\leq H< 1/2$, we construct entire $H$-graphs in $\mathbb{H}^2\times\mathbb{R}$ that are parabolic and not invariant by one parameter groups of isometries of $\mathbb{H}^2\times\mathbb{R}$. Their asymptotic boundaries are $(\partial_\infty\mathbb{H}^2)\times\mathbb{R}$; they are dense at infinity. When $H=0$ the examples are minimal graphs constructed by P. Collin and the second author [2].

Group of Isometries of Hilbert Ball Equipped with the Carathéodory Metric

In this article, we study the geometry of an infinite-dimensional hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carathéodory metric and learn about some special subclasses of this group. We will also find some unitary equivalence condition and compute some cardinalities.

Isomorphism between Spaces of Multilinear Maps and Nested Compositions over Real Normed Vector Spaces

Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.

Corporate internet reporting in Egypt: a pre- and peri-uprising analysis

PurposeThe purpose of this study is to provide a detailed descriptive account and analysis of corporate internet reporting (CIR) practices among non-financial companies listed on the Egyptian Exchange (EGX) at two points in time – December 2010 (pre) and December 2013 (peri) political and social unrest in Egypt.Design/methodology/approachThe study developed a disclosure index to determine the extent of CIR practices among all non-financial companies listed on the EGX in December 2010 and December 2013. The study uses ordinary least squares (OLS) regressions and isometric log-ratio transformations for compositional independent variables to empirically examine the factors affecting CIR in Egypt using a modern institutional theory lens.FindingsThe findings of this investigation suggest that listed companies in Egypt have started embracing the power of the internet as a disclosure channel, but the extent of these practices increased significantly over the investigated period, with great variations evident among the sampled companies in this regard. Such variations were chiefly dependent on the changing institutional actors over the two time frames. Additionally, the findings show that the time factor is particularly important for a given institutional field to induce a sufficient diffusion of corporate practices, especially in periods with drastic institutional change.Practical implicationsThe evidence presented reflects the voluntary nature of CIR practices and the absence of a reinforced regulatory framework for organizing and monitoring such practices, with companies having discretion in terms of the amount and type of information disclosed via their websites. The results should, therefore, provide useful guidelines for regulators and standard-setters in identifying best practices, which, in turn, should allow CIR practices to become more consistent, making them easier to monitor and govern.Originality/valueTo the best of the authors’ knowledge, this is the first study that examines CIR practices at two points in time using a comprehensive disclosure index and a modern institutional theory lens.

Connected components of Isom(varvec{pmb {mathbb {H}}^3})-representations of non-orientable surfaces

Let N_k denote the closed non-orientable surface of genus k. In this paper we study the behaviour of the ‘square map’ from the group of isometries of hyperbolic 3-space to the subgroup of orientation preserving isometries. The properties of the ‘square map’ and other related maps serve as a technical step towards the counting of the connected components of the variety of representations of pi _1(N_k) in Isom(mathbb {H}^3). We show that the variety of representations hom(pi _1(N_k),mathrm {Isom}(mathbb {H}^3)) has 2^{k+1} connected components, which are distinguished by the Stiefel-Whitney classes of the associated flat bundle.

Non‐commutative amoebas

The group of isometries of the hyperbolic space H 3 $ \mathbb {H}^3$ is the 3-dimensional group PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ , which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO ( 3 ) ⊂ PSL 2 ( C ) $\operatorname{SO}(3)\subset \operatorname{PSL}_2(\mathbb {C})$ naturally maps it back to H 3 $ \mathbb {H}^3$ . Each fiber of this map is diffeomorphic to the real projective 3-space R P 3 ${\mathbb {R}}{\mathbb {P}}^3$ . The resulting map PSL 2 ( C ) → H 3 $\operatorname{PSL}_2(\mathbb {C})\rightarrow \mathbb {H}^3$ can be viewed as the simplest non-commutative counterpart of the map Log : ( C × ) n → R n $\operatorname{Log}:(\mathbb {C}^\times )^n\rightarrow \mathbb {R}^n$ from the commutative complex Lie group ( C × ) n $(\mathbb {C}^\times )^n$ with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties V ⊂ ( C × ) n $V\subset (\mathbb {C}^\times )^n$ as images Log ( V ) ⊂ R n $\operatorname{Log}(V)\subset \mathbb {R}^n$ . We define the amoeba of an algebraic subvariety of PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ as its image in H 3 $ \mathbb {H}^3$ . The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas R n $\mathbb {R}^n$ .

Spacetimes with continuous local isotropies

General relativistic spacetimes in which at every point [Formula: see text] in some neighbourhood [Formula: see text] the tangent space [Formula: see text] is invariant under the same continuous subgroup [Formula: see text] of the Lorentz group are considered. Some previously open problems for such spacetimes are discussed and solved, and a comprehensive survey of such spacetimes is provided. For most cases, invariance of the curvature and its first covariant derivative under [Formula: see text] imply the existence of an isometry group in [Formula: see text] containing [Formula: see text] as an isotropy group. In the remaining cases, invariance of second and third derivatives may be required for this implication, and in particular, the necessity of invariance of the third derivative is proved for the locally rotationally symmetric spacetimes discussed by Ellis and by Stewart and Ellis.

Classification of the Orthogonal Separable Webs for the Hamilton-Jacobi and Klein-Gordon Equations on 3-Dimensional Minkowski Space

We review a new theory of orthogonal separation of variables on pseudo-Riemannian spaces of constant zero curvature via concircular tensors and warped products. We then apply this theory to three-dimensional Minkowski space, obtaining an invariant classification of the forty-five orthogonal separable webs modulo the action of the isometry group. The eighty-eight inequivalent coordinate charts adapted to the webs are also determined and listed. We find a number of separable webs which do not appear in previous works in the literature. Further, the method used seems to be more efficient and concise than those employed in earlier works.

The Poincaré Exponent and the Dimensions of Kleinian Limit Sets

Kleinian groups are discrete groups of isometries of hyperbolic space. Their actions give rise to intricate fractal limit sets on the boundary at infinity and there is great interest in estimating the “dimension” of these limit sets. As an invitation to this fascinating area, we provide a proof of the (well-known) result that the Poincaré exponent of a nonelementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof uses only elementary hyperbolic and fractal geometry.

Isometry groups of three-dimensional Lie groups

We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism are contained in the subspace of classes of metrics with maximal index of symmetry.

Random walks on hyperbolic spaces: Concentration inequalities and probabilistic Tits alternative

The goal of this article is two-fold: in a first part, we prove Azuma–Hoeffding type concentration inequalities around the drift for the displacement of non-elementary random walks on hyperbolic spaces. For a proper hyperbolic space M, we obtain explicit bounds that depend only on M, the size of support of the measure as in the classical case of sums of independent random variables, and on the norm of the driving probability measure in the left regular representation of the group of isometries. We obtain uniform bounds in the case of hyperbolic groups and effective bounds for simple linear groups of rank-one. In a second part, using our concentration inequalities, we give quantitative finite-time estimates on the probability that two independent random walks on the isometry group of a hyperbolic space generate a free non-abelian subgroup. Our concentration results follow from a more general, but less explicit statement that we prove for cocycles which satisfy a certain cohomological equation. For example, this also allows us to obtain subgaussian concentration bounds around the top Lyapunov exponent of random matrix products in arbitrary dimension.

Orthogonal ℓ1-sets and extreme non-Arens regularity of preduals of von Neumann algebras

A Banach algebra A is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols when A⁎=WAP(A). To identify the opposite behaviour, Granirer called a Banach algebra extremely non-Arens regular (enAr, for short) when the quotient A⁎/WAP(A) contains a closed subspace that has A⁎ as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra A is r-enAr, with r≥1, when there is an isomorphism with distortion r of A⁎ into A⁎/WAP(A). When r=1, we obtain an isometric isomorphism and we say that A is isometrically enAr. We then identify sufficient conditions for the predual V⁎ of a von Neumann algebra V to be r-enAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be r-enAr:(i)the weighted semigroup algebra of any weakly cancellative discrete semigroup, when the weight is diagonally bounded with diagonal bound c≥r. When the weight is multiplicative, i.e., when c=1, the algebra is isometrically enAr,(ii)the weighted group algebra of any non-discrete locally compact infinite group and for any weight,(iii)the weighted measure algebra of any locally compact infinite group, when the weight is diagonally bounded with diagonal bound c≥r. When the weight is multiplicative, i.e., when c=1, the algebra is isometrically enAr. The Fourier algebra A(G) of a locally compact infinite group G is shown to be isometrically enAr provided that (1) the local weight of G is greater or equal than its compact covering number, or (2) G is countable and contains an infinite amenable subgroup.

A note on Selberg’s lemma and negatively curved Hadamard manifolds

Author(s): Kapovich, Michael | Abstract: Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.

A new non-inheriting homogeneous solution of the Einstein-Maxwell equations with cosmological term

We find a new homogeneous solution to the Einstein-Maxwell equations with a cosmological term. The spacetime manifold is $R \times S^3$. The spacetime metric admits a simply transitive isometry group $G = R \times SU(2)$ of isometries and is of Petrov type I. The spacetime is geodesically complete and globally hyperbolic. The electromagnetic field is non-null and non-inheriting: it is only invariant with respect to the $SU(2)$ subgroup and is time-dependent in a stationary reference frame.

The Effects of Non-Digital Game-Based Learning and Cognitive Level of Questions on Isometric Transformations

Abstract: The present study examined the effects of the non-digital game-based instructional method and the cognitive level of questions in determining students’ achievement in mathematics, particularly in Isometric Transformation. The study used a quasi-experimental design and involved 116 participants (i.e., form two students) from two local secondary schools in Kuching. The participants were divided into control and experimental groups – the control group learned the Isometric Transformation topic via conventional learning, while the experimental group used non-digital game-based learning (NDGBL). The results showed that participants who learned using the NDGBL scored better in the Isometric Transformation test than those who learned the topic using the conventional instructional method. The present findings also showed that participants’ achievement of Isometric Transformation differed significantly between cognitive levels of questions. In particular, participants’ achievements were recorded better for lower-level cognitive questions than higher-level cognitive questions. However, there was no significant interaction effect between the instructional method and the cognitive levels of questions on participants’ achievement in the topic learned. The study provided empirical evidence on the role of NDGBL in learning Isometric Transformation, in that it should be considered as an alternative approach for learning Isometric Transformation and Mathematics in general, with an effective integration into the secondary mathematics curriculum.&#x0D; &#x0D; Keywords: Game-Based Learning, Conventional Learning, Isometric Transformations, Students’ Achievement, Thinking Skills

Strong morita equivalence for conditional expectations

AbstractWe consider two inclusions of $C^{*}$-algebras whose small $C^{*}$-algebras have approximate units of the large $C^{*}$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^{*}$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study the basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss the Picard group for a bimodule linear map.

Symmetries of exotic negatively curved manifolds

Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. $\operatorname{Isom}(M)$ acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, the order of $\operatorname{Isom}(M)$ can be arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of $\operatorname{Isom}(M)$ of “small” index acts by diffeomorphisms of $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky–Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

Growth of mod-2 homology in higher-rank locally symmetric spaces

Let X be a higher-rank symmetric space or a Bruhat–Tits building of dimension at least 2 such that the isometry group of X has property (T). We prove that for every torsion-free lattice Γ⊂IsomX, any homology class in H1(Γ∖X,F2) admits a representative cycle of total length oX(Vol(Γ∖X)). As an application, we show that dimF2H1(Γ∖X,F2)=oX(Vol(Γ∖X)).