Isometry Research Articles

A “magic” approach to octonionic Rosenfeld spaces

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find seven pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation á la Rosenfeld.

A de Sitter no-hair theorem for 3+1d cosmologies with isometry group forming 2-dimensional orbits

We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations.

A remark on convex functions and isometry groups of Lorentzian manifolds

A remark on convex functions and isometry groups of Lorentzian manifolds

Dimension of the isometry group in spacetimes with an invariant frame

The necessary and sufficient conditions for a spacetime with an invariant frame to admit a group of isometries of dimension r are given in terms of the connection tensor H associated with this frame. In Petrov–Bel types I, II and III, and in other spacetimes where an invariant frame algebraically defined by the curvature tensor exists, the connection tensor H is given in terms of the Weyl and Ricci tensors without an explicit determination of the frame. Thus, an intrinsic, deductive, explicit and algorithmic characterization of these spacetimes follows. Some examples show that this algorithm can be easily implemented on the xAct Mathematica suite of packages.

Dixon-Rosenfeld lines and the Standard Model

We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra mathbb {C}otimes mathbb {H}otimes mathbb {O}. Three different Lie groups are found as isometry groups of these coset manifolds using Tits’ formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal–Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within mathbb {C}otimes mathbb {H}otimes J_{2}(mathbb {O}), mathbb {O}otimes J_{2}(mathbb {C}otimes mathbb {H}), and mathbb {C}otimes mathbb {O}otimes J_{2}(mathbb {H}).

A Class of Locally Inhomogeneous Complete Quaternionic Kähler Manifolds

AbstractWe prove that the one-loop deformation of any quaternionic Kähler manifold in the class of c-map spaces is locally inhomogeneous. As a corollary, we obtain that the full isometry group of the one-loop deformation of any homogeneous c-map space has precisely cohomogeneity one.

Circle companions of Hardy spaces of the unit disk

This paper gives a complete answer to the following problem: Find the circle companion of the Hardy space of the unit disk with values in the space of all bounded linear operators between two separable Hilbert spaces. Classically, the problem asks whether for each function h on the unit disk, there exists a “boundary function” bh on the unit circle such that the mapping bh↦h is an isometric isomorphism between Hardy spaces of the unit circle and the unit disk with values in some Banach space. For the case of bounded linear operator-valued functions, we construct a Hardy space of the unit circle such that its elements are SOT measurable, and their norms are integrable: indeed, this new space is isometrically isomorphic to the Hardy space of the unit disk via a “strong Poisson integral.”

Regularization of electromagnetic field for self-force problem in de Sitter spacetime

The paper is concerned with the motion of a point electric charge in de Sitter spacetime. A point particle of mass m and charge q moving on a geodesic curve produces electromagnetic field that diverges at a particle’s position. The field is determined by the electromagnetic Green’s function by Higuchi and Lee (2008 Phys. Rev. D 78 084031). The self-force contains both divergent and finite terms, and the latter are responsible for the radiation reaction. Our derivation of an effective equations of motion is based on conservation laws corresponding to the group of isometry of de Sitter space. The Nöther quantities consist of particle’s individual characteristics and energy, momentum, and angular momentum carried by particle’s electromagnetic field. Following the Detweiler–Whiting concept that a charge’s motion should only be enforced by the regular component of its own field, we ignore the divergent terms in conservation laws. We assume that the divergencies are absorbed by particle’s individual characteristics within the renormalization procedure. Finite radiative terms together with kinematic particle’s characteristics constitute ten conserved quantities of closed particle plus field system. Their differential consequences yield the effective equations of motion of radiating charge in an external electromagnetic field and gravitation. Contributions to already renormalized particle’s four-momentum and its inertial mass originated from electromagnetic field and background gravity are also derived from ten balance equations.

On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface

Let X denote a K3 surface over an arbitrary field k. Let ks denote a separable closure of k and let Xs denote the base change of X to ks. Let O(PicX) and O(PicXs) denote the group of isometries of the lattices PicX and PicXs, respectively. Let RX denote the Galois invariant part of the Weyl group of PicXs. One can show that each element in RX can be restricted to an element of O(PicX). The following question arises: Is the image of the restriction mapRX→O(PicX)a normal subgroup of O(PicX)for every K3 surface X? We show that the answer is negative by giving counterexamples over k=Q.

Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces

We study random walks on the isometry group of a Gromov hyperbolic space or Teichmüller space. We prove that the translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment. While doing this, we recover the central limit theorem of Benoist and Quint for the displacement of a reference point and establish its converse. Also discussed are the corresponding laws of the iterated logarithm. Finally, we prove sublinear geodesic tracking by random walks with finite (1/2)-th moment and logarithmic tracking by random walks with finite exponential moment.

Aitchison’s Compositional Data Analysis 40 Years on: A Reappraisal

The development of John Aitchison’s approach to compositional data analysis is followed since his paper read to the Royal Statistical Society in 1982. Aitchison’s logratio approach, which was proposed to solve the problematic aspects of working with data with a fixed-sum constraint, is summarized and reappraised. It is maintained that the properties on which this approach was originally built, the main one being subcompositional coherence, are not required to be satisfied exactly—quasi-coherence is sufficient, that is near enough to being coherent for all practical purposes. This opens up the field to using simpler data transformations, such as power transformations, that permit zero values in the data. The additional property of exact isometry, which was subsequently introduced and not in Aitchison’s original conception, imposed the use of isometric logratio transformations, but these are complicated and problematic to interpret, involving ratios of geometric means. If this property is regarded as important in certain analytical contexts, for example, unsupervised learning, it can be relaxed by showing that regular pairwise logratios, as well as the alternative quasi-coherent transformations, can also be quasi-isometric, meaning they are close enough to exact isometry for all practical purposes. It is concluded that the isometric and related logratio transformations such as pivot logratios are not a prerequisite for good practice, although many authors insist on their obligatory use. This conclusion is fully supported here by case studies in geochemistry and in genomics, where the good performance is demonstrated of pairwise logratios, as originally proposed by Aitchison, or Box–Cox power transforms of the original compositions where no zero replacements are necessary.

Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations

An essential space curve in the study of differential geometry is the rectifying curve. In this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. The normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the Christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. Moreover, we also studied some properties for the first fundamental form of the surfaces.

Equivariant index bound for min–max free boundary minimal surfaces

Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min–max procedure á la Simon–Smith with n-parameters is bounded above by n.

Geometric and isometric properties of two classes of sequence spaces

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X n X_n , the completion of c 00 c_{00} relatively to the norm \[ ‖ x ‖ n = sup { ∑ i ∈ F | x i | , with , F of cardinality at most n } . \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X n X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T [ θ , A n ] T[\theta , A_n] ( 0 > θ > 1 0>\theta >1 ) associated with the regular family A n A_n , comprised of all subsets of N \mathbb {N} with at most n n elements. For this space, we characterize the extreme points and show that for θ ≤ 1 n \theta \leq \frac {1}{n} , T [ θ , A n ] T[\theta , A_n] supports standard isometries. We also derive the form for the surjective isometries of T [ θ , A 3 ] T[\theta , A_3] .

Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

AbstractIn this paper, we study the groups of isometries and the set of bi‐Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

Primary Schoolers' Response to a Multisensory Serious Game on Cartesian Plane Coordinates in Immersive Virtual Reality.

The Cartesian coordinate system is a fundamental concept for mathematics and science and poses a teaching challenge at primary school level. Learning the Cartesian coordinate system has the potential to promote numerical cognition through number-space associations, as well as core geometric concepts, including isometric transformations, symmetry, and shape perception. Immersive virtual reality (VR) facilitates embodied forms of teaching and learning mathematics through whole-body sensorimotor interaction and offers benefits as a platform to learn the Cartesian coordinate system compared with "real world" classroom activities. Our goal was to validate the Cartesian-Garden, a serious game designed to provide an educationally robust but engaging vehicle to teach these concepts in primary-level mathematics in a multisensory VR environment. In the game, the child explores a Cartesian-Garden, that is, a field of flowers in which each flower corresponds to x and y coordinates. Specifically, we tested whether exploring numbers spatially represented improved spatial and numerical skills independently from the use of VR. Children (n = 49; age 7-11 years old) were divided into experimental and age-matched control groups. The experimental group explored the Cartesian-Garden and picked flowers corresponding to target coordinates; the control group played a VR game unrelated to Cartesian coordinates. To quantify potential improvements, children were tested before and after training with perceptual tests investigating number line and spatial thinking. The results point toward differential age-related improvements depending on the tested concept, especially for the number line. This study provides the guidelines for the successful use of the Cartesian-Garden game, beneficial for specific age groups.

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D′(X) forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R(z) extends, and the residue representations can be infinite-dimensional.

Group Actions on Twisted Sums of Banach Spaces

We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of G-centralizer and G-equivariant map. We will show that when (A) G is an amenable group and (U) the target space is complemented in its bidual by a G-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of G-centralizers; and that, under (A) and (U), G-centralizers are bounded perturbations of G-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of L_p(0,1), pne 2 on the Kalton–Peck space Z_p, certain non-unitarizable triangular representations of the free group {mathbb {F}}_infty on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of L_p-spaces. In the penultimate section we consider the category of G-Banach spaces and study its exact sequences, showing that, under (A) and (U), G-splitting and usual splitting coincide. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.

Planar In-Hand Manipulation Using Primitive Rotations Based on Isometric Transformations

Achieving in-hand manipulation of a grasped object is challenging in robotic communities. This article proposes a planar intrinsic in-hand manipulation approach based on isometric transformation. We demonstrate that the isometric transformation (excluding reflections) can be reduced to rotations with two fixed centers, if more than two rotations around two different centers are performed. Furthermore, this article applies this theory to reposition and reorient a grasped object by performing three or more primitive rotations sequentially. We further analyze the feasible manipulation space, where the desired position and orientation of a grasped object can be attained. The theoretical results have been verified through simulations and experiments. In addition, we performed several real-world tasks using a novel robotic hand with two rotational degrees of freedom at each finger, demonstrating that the proposed in-hand manipulation method can reposition and reorient different grasped objects in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{SE}(2)$</tex-math></inline-formula> .

VSP-Fuse: Multifocus Image Fusion Model Using the Knowledge Transferred From Visual Salience Priors

Multifocus image fusion (MFIF), as an efficient way to improve the visual effect of images with partial focus defects, is of great significance in the field of image enhancement. According to the imaging principle of the lens, we summarize the visual salience priors (VSP) from the daily photo scene and two relationships from MFIF. Thereby, an edge-sensitive model for MFIF is presented in this study. Supported by VSP, we consider the correlation between salience object detection (SOD) and MFIF, and select the former as a pre-training task. SOD provides the network with realistic depth of field and bokeh effects to learn, and enhances the network’s ability to extract and express the edges of focused objects. Meanwhile, given the scarcity of real multifocus training sets, we propose a randomized approach to generate massive training sets and pseudo-labels based on limited unlabeled data. Besides, two attention modules are designed based on isometric domain transformation (IDT) in the traditional edge-preservation field. IDT removes interference information from feature maps in a low-cost manner, thereby facilitating channel-wise and spatial-wise weight assignments. Experimental results on four datasets show that the performance of our model is superior to that of many supervised models, without the need of any real MFIF training set.