Isometry Research Articles

Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F = ( F 1 , … , F m ) {F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹 n {\mathbb{B}^{n}} , n ≥ 2 {n\geq 2} , to a bounded symmetric domain Ω = Ω 1 × ⋯ × Ω m {\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ω i ′ {\Omega_{i}^{\prime}} s are irreducible factors of Ω. We prove every non-constant component F i {F_{i}} must map generic boundary points of 𝔹 n {\mathbb{B}^{n}} to the boundary of Ω i {\Omega_{i}} . In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls.

On Finite Homogeneous Metric Spaces

This survey is devoted to recently obtained results on finite homogeneousmetric spaces. The main subject of discussion is the classification of regular and semiregular polytopes in Euclidean spacesby whether or not their vertex sets have the normal homogeneity property or the Clifford - Wolf homogeneity property.Every finite homogeneous metric subspace of an Euclidean space represents the vertex set of a compact convex polytope with the isometry group that is transitive on the set of vertices, moreover, all these vertices lie on some sphere. Consequently, the study of such subsets is closely related to the theory of convex polytopes in Euclidean spaces. The normal generalized homogeneity and the Clifford - Wolf homogeneity describe more stronger properties than the homogeneity. Therefore, it is natural to first check the presence of these properties for the vertex sets of regular and semiregular polytopes. In addition to the classification results, the paper contains a description of the main tools for the study of the relevant objects.

Conjugation matters. Bioctonionic Veronese vectors and Cayley–Rosenfeld planes

Motivated by the recent interest in Lie algebraic and geometric structures arising from tensor products of division algebras and their relevance to high energy theoretical physics, we analyze generalized bioctonionic projective and hyperbolic planes. After giving a Veronese representation of the complexification of the Cayley plane [Formula: see text], we present a novel, explicit construction of the bioctonionic Cayley–Rosenfeld plane [Formula: see text], again by exploiting Veronese coordinates. We discuss the isometry groups of all generalized bioctonionic planes, recovering all complex and real forms of the exceptional groups [Formula: see text] and [Formula: see text], and characterizing such planes as symmetric and Hermitian symmetric spaces. We conclude by discussing some possible physical applications.

Trivializable and Quaternionic Subriemannian Structures on {mathbb {S}}^7 and Subelliptic Heat Kernel

On the seven dimensional Euclidean sphere {mathbb {S}}^7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution {mathcal {H}}_T is induced by a Clifford module structure of {mathbb {R}}^8. More precisely, {mathcal {H}}_T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution {mathcal {H}}_{Q} of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration {mathbb {S}}^3hookrightarrow {mathbb {S}}^7rightarrow {mathbb {S}}^4. Answering a question in: Markina and Godoy Molina (Rev Mat Iberoam 27(3), 997–1022, 2011) we first show that {mathcal {H}}_{Q} does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures [20], the intrinsic sublaplacians Delta _{text {sub}}^T and Delta _{text {sub}}^{Q} and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

On Finslerian extension of special relativity

In this paper, we demonstrate that Robb-Geroch’s definition of a relativistic interval admits a simple and fairly natural generalization leading to a Finsler extension of special relativity. Another justification for such an extension goes back to the works of Lalan and Alway and, finally, was put on a solid basis and systematically investigated by Bogoslovsky under the name “Special-relativistic theory of locally anisotropic spacetime”. The isometry group of this spacetime, DISIM[Formula: see text], is a deformation of the Cohen and Glashow’s very special relativity symmetry group ISIM(2). Thus, the deformation parameter [Formula: see text] can be regarded as an analog of the cosmological constant characterizing the deformation of the Poincaré group into the de Sitter (anti-de Sitter) group. The simplicity and naturalness of Finslerian extension in the context of this paper adds weight to the argument that the possibility of a nonzero value of [Formula: see text] should be carefully considered.

S-Duality and the Universal Isometries of q-Map Spaces

The tree-level q-map assigns to a projective special real (PSR) manifold of dimension n-1ge 0, a quaternionic Kähler (QK) manifold of dimension 4n+4. It is known that the resulting QK manifold admits a (3n+5)-dimensional universal group of isometries (i.e. independently of the choice of PSR manifold). On the other hand, in the context of Calabi–Yau compactifications of type IIB string theory, the classical hypermultiplet moduli space metric is an instance of a tree-level q-map space, and it is known from the physics literature that such a metric has an mathrm {SL}(2,{mathbb {R}}) group of isometries related to the mathrm {SL}(2,{mathbb {Z}}) S-duality symmetry of the full 10d theory. We present a purely mathematical proof that any tree-level q-map space admits such an mathrm {SL}(2,{mathbb {R}}) action by isometries, enlarging the previous universal group of isometries to a (3n+6)-dimensional group G. As part of this analysis, we describe how the (3n+5)-dimensional subgroup interacts with the mathrm {SL}(2,{mathbb {R}})-action, and find a codimension one normal subgroup of G that is unimodular. By taking a quotient with respect to a lattice in the unimodular group, we obtain a quaternionic Kähler manifold fibering over a projective special real manifold with fibers of finite volume, and compute the volume as a function of the base. We furthermore provide a mathematical treatment of results from the physics literature concerning the twistor space of the tree-level q-map space and the holomorphic lift of the (3n+6)-dimensional group of universal isometries to the twistor space.

Inertial-Based Integration With Transformed INS Mechanization in Earth Frame

This article proposes to use a newly derived transformed inertial navigation system (INS) mechanization to fuse INS with other complementary navigation systems. Through formulating the attitude, velocity, and position as one group state of a group of double direct spatial isometries <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb{S}{\mathbb{E}_2}(3)$</tex-math></inline-formula> , the transformed INS mechanization has proven to be group affine, which means that the corresponding vector error state model will be trajectory-independent. In order to make use of the transformed INS mechanization in inertial-based integration, both the right and left vector error state models are derived. The INS/GPS and INS/Odometer integration are investigated as two representatives of inertial-based integration. Some application aspects of the derived error state models in the two applications are presented, which include how to select the error state model, initialization of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb{S}{\mathbb{E}_2}(3)$</tex-math></inline-formula> based error state covariance, and feedback correction corresponding to the error state definitions. Extensive Monte Carlo simulations and land vehicle experiments are conducted to evaluate the performance of the derived error state models. It is shown that the most striking superiority of using the derived error state models is their ability to handle the large initial attitude misalignments, which is just the result of log-linearity property of the derived error state models. Therefore, the derived error state models can be used in the so-called attitude alignment for the two applications. Moreover, the derived right error state-space model is also very preferred for long-endurance INS/Odometer integration due to the filtering consistency caused by its less dependence on the global state estimate.

Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean, or Lorentz, or generally Isotropic Metric

Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean, or Lorentz, or generally Isotropic Metric

Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design.

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a manifold equipped with an indefinite metric. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this representation - independent of the network - with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

Dynamical realizations of the Lifshitz group

Dynamical realizations of the Lifshitz group are studied within the group-theoretic framework. A generalization of the 1d conformal mechanics is constructed, which involves an arbitrary dynamical exponent z. A similar generalization of the Ermakov-Milne-Pinney equation is proposed. Invariant derivative and field combinations are introduced, which enable one to construct a plethora of dynamical systems enjoying the Lifshitz symmetry. A metric of the Lorentzian signature in (d+2)-dimensional spacetime and the energy-momentum tensor are constructed, which lead to the generalized Ermakov-Milne-Pinney equation upon imposing the Einstein equations. The method of nonlinear realizations is used for building Lorentzian metrics with the Lifshitz isometry group. In particular, a (2d+2)-dimensional metric is constructed, which enjoys an extra invariance under the Galilei boosts.

Curvatures and Austere Property of Orbits of Path Group Actions Induced by Hermann Actions

It is known that an isometric action of a Lie group on a compact symmetric space gives rise to a proper Fredholm action of a path group on a path space via the gauge transformations. In this paper, supposing that the isometric action is a Hermann action (i.e. an isometric action of a symmetric subgroup of the isometry group) we give an explicit formula for the principal curvatures of orbits of the path group action and study the condition for those orbits to be austere, that is, the set of principal curvatures in the direction of each normal vector is invariant under the multiplication by minus one. To prove the results we essentially use the facts that Hermann actions are hyperpolar and all orbits of Hermann actions are curvature-adapted submanifolds. The results greatly extend the author's previous result in the case of the standard sphere and show that there exist a larger number of infinite dimensional austere submanifolds in Hilbert spaces.

Geometry of Gyrogroups via Klein’s Approach

Using Klein's approach, geometry can be studied in terms of a space of points and a group of transformations of that space. This allows us to apply algebraic tools in studying geometry of mathematical structures. In this article, we follow Klein's approach to study the geometry $(G, \mathcal{T})$, where $G$ is an abstract gyrogroup and $\mathcal{T}$ is an appropriate group of transformations containing all gyroautomorphisms of $G$. We focus on $n$-transitivity of gyrogroups and also give a few characterizations of coset spaces to be minimally invariant sets. We then prove that the collection of open balls of equal radius is a minimally invariant set of the geometry $(G, \Gamma_m)$ for any normed gyrogroup $G$, where $\Gamma_m$ is a suitable group of isometries of $G$.

On local isometries between algebras of C(Y)-valued differentiable maps

Let K be either the real unit interval [0, 1] or the complex unit circle {mathbb {T}} and let C(Y) be the space of all complex-valued continuous functions on a compact Hausdorff space Y. We prove that the isometry group of the algebra C^1(K,C(Y)) of all C(Y)-valued continuously differentiable maps on K, equipped with the Sigma -norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive.

Gaussian free fields and Riemannian rigidity

In the present paper, we show that on a compact Riemannian manifold $(M,g)$ of dimension $d\leqslant 4$ whose metric has negative curvature, the renormalized partition function $Z_g(\lambda)$ of a massive Gaussian Free Field determines the length spectrum of $(M,g)$ and imposes some strong geometric constraints on the Riemannian structure of $(M,g)$. In any finite dimensional family of Riemannian metrics of negative sectional curvature bounded from below and above and whose isometry group is trivial, there is only a \textbf{finite number of isometry classes} of metrics with given partition function $Z_g(\lambda)$. When $d<4$, the same result holds true if the random variable $\int_M:\phi^2:dv$ has given probability distribution and without the lower bound on the sectional curvatures.

A Wavelet Plancherel Theory with Application to Multipliers and Sparse Approximations

We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space – the so called window–signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window–signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window–signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window–signal space. As one example application, we show how continuous wavelet multipliers (also called Calderón–Toeplitz operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.

Topology-based representative datasets to reduce neural network training resources

One of the main drawbacks of the practical use of neural networks is the long time required in the training process. Such a training process consists of an iterative change of parameters trying to minimize a loss function. These changes are driven by a dataset, which can be seen as a set of labeled points in an n-dimensional space. In this paper, we explore the concept of a representative dataset which is a dataset smaller than the original one, satisfying a nearness condition independent of isometric transformations. Representativeness is measured using persistence diagrams (a computational topology tool) due to its computational efficiency. We theoretically prove that the accuracy of a perceptron evaluated on the original dataset coincides with the accuracy of the neural network evaluated on the representative dataset when the neural network architecture is a perceptron, the loss function is the mean squared error, and certain conditions on the representativeness of the dataset are imposed. These theoretical results accompanied by experimentation open a door to reducing the size of the dataset to gain time in the training process of any neural network.

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space {mathbb {E}}^{3}(kappa ,tau ) with isometry group of dimension 4 is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satisfying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in {mathbb {E}}^3(kappa ,tau ), becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

Isometric rigidity of Wasserstein spaces: The graph metric case

The aim of this paper is to prove that the p p -Wasserstein space W p ( X ) \mathcal {W}_p(X) is isometrically rigid for all p ≥ 1 p\geq 1 whenever X X is a countable graph metric space. As a consequence, we obtain that for every countable group H {H} and any p ≥ 1 p\geq 1 there exists a p p -Wasserstein space whose isometry group is isomorphic to H {H} .

A Compositional Analysis of Physical Activity, Sedentary Time, and Sleep and Associated Health Outcomes in Children and Adults with Cystic Fibrosis.

This study sought to investigate the association of light physical activity (LPA), moderate-to-vigorous physical activity (MVPA), sedentary time (SED), and sleep with lung function in children and adults with CF. In total, 86 children (41 females; 13.6 ± 2.8 years; FEV1%predicted: 86 ± 1%) and 43 adults (21 females; 24.6 ± 4.7 years; FEV1%predicted: 63 ± 21%) with CF participated in this study. Wrist-worn accelerometery was used to assess PA, SED and sleep. Compositional linear regression models were conducted following normalisation via isometric log-ratio transformations. Sequential binary partitioning was applied to investigate the impact of reallocating 10 to 30 min between each behaviour on FEV1%predicted. A decline in FEV1%predicted was predicted with the reallocation of 30 min from MVPA to SED or LPA or sleep to any other behaviour in children (−3.04–−0.005%) and adults (−3.58–−0.005%). Conversely, improvements in FEV1%predicted were predicted when 30 min was reallocated to MVPA from LPA or SED in children (0.12–1.59%) and adults (0.77–2.10%), or when 30 min was reallocated to sleep from any other behaviour in both children (0.23–2.56%) and adults (1.08–3.58%). This study supports the importance of MVPA and sleep for maintaining and promoting lung function in people with CF.

Design of pre-stressed plate-strips to cover non-developable shells

In this paper we address the following design problem: what is the shape of a plate and the associated pre-stress that relaxes toward a given three-dimensional shell ? As isometric transformations conserve the gaussian curvature, three-dimensional non-developable shells cannot be obtained from the relaxation of pre-strained plates by using isometric transformations only. Overcoming this geometric restriction, including small-strains and large rotations, solves the problem for small areas only. This paper dispenses with the small-area restriction to cover three-dimensional shells fully by using shell-strips. Since shell-strips have an additional geometric parameter, we show that under suitable assumptions that relate the width of the strip to the curvature of the shell, we are able to design arbitrary shell surfaces by covering them with shell-strips. As an illustration, we provide optimized covers of the sphere in a variety of different surface-strips relaxed from plate-strips with homogeneous and isotropic pre-stress. Moreover, we propose the design of the torus, of the helicoid and of the non-developable Möbius band, which requires inhomogeneous and anisotropic pre-stress.