Symmetric Manifolds Research Articles

Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras

Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras

Lie PCA: Density estimation for symmetric manifolds

We introduce an extension to local principal component analysis for learning symmetric manifolds. In particular, we use a spectral method to approximate the Lie algebra corresponding to the symmetry group of the underlying manifold. We derive the sample complexity of our method for a variety of manifolds before applying it to various data sets for improved density estimation.

On the Symmetries of a Kaehler Manifold

The natural symmetries of Riemannian manifolds are described by the symmetries of its Riemann curvature tensor. In that sense, the most symmetric manifolds are the constant sectional curvature ones. Its natural generalizations are locally symmetric manifolds, semisymmetric manifolds, and pseudosymmetric manifolds. The analogous generalizations of constant holomorphic sectional curvature Kaehler manifolds are locally symmetric Kaehler manifolds, semisymmetric Kaehler manifolds, and holomorphically pseudosymmetric Kaehler manifolds. Do they differ in some way from their Riemannian analogues? Yes, we prove they can all be characterized only in terms of holomorphic planes. Furthermore, the concept of holomorphically pseudosymmetric Kaehler manifold is different from the classical notion of pseudosymmetric Riemannian manifold proposed by Deszcz. We study some relations between both definitions of pseudosymmetry and the so called double sectional curvatures in the sense of Deszcz. We also present a geometric interpretation of the complex Tachibana tensor and a new characterization of constant holomorphic sectional curvature Kaehler manifolds.

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.

Subscripto multiplex: A Riemannian symmetric positive definite strategy for offline signature verification

The human handwritten signature is considered to be a significant biometric trait. In the case of offline signatures, the problem is addressed as an image recognition task. On the other hand, the visual representation of symmetric positive definitive matrices, usually by means of the covariance descriptor of the image feature maps, forms a specific Riemannian manifold with a widespread usage and a favorable performance in a plethora of applications. Surprisingly, no records of offline-signature-verification-oriented research in the space of symmetric positive definitive matrix have been found up to now. In this work, we propose, for the first time in offline signature-verification literature, mapping of handwritten signature images in points of the tangent space of a connected symmetric positive definitive manifold for verification purposes. Furthermore, based on the principles of differential geometry, we address the notorious limited training problem of offline signature verification in this manifold by proposing two different feature augmentation methods. The efficiency of the proposed method is evaluated using three popular datasets of Western and Asian origin. Error rates against skilled and random forgery in both baselines as well augmentation scenarios are strong indicators of the informative and highly discriminative nature of symmetric positive definitive manifold oriented representation.

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here, we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $$M = G/H$$ such that a nilpotent analytic subgroup of G is transitive on M. Suppose that there is a reductive decomposition $${\mathfrak {g}}= {\mathfrak {h}}\oplus {\mathfrak {n}}$$ (vector space direct sum) with $${\mathfrak {n}}$$ nilpotent. When the metric is nondegenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case, we construct examples to show that the number of nilpotency steps is unbounded.

Learning to Optimize on Riemannian Manifolds.

Many learning tasks are modeled as optimization problems with nonlinear constraints, such as principal component analysis and fitting a Gaussian mixture model. A popular way to solve such problems is resorting to Riemannian optimization algorithms, which yet heavily rely on both human involvement and expert knowledge about Riemannian manifolds. In this paper, we propose a Riemannian meta-optimization method to automatically learn a Riemannian optimizer. We parameterize the Riemannian optimizer by a novel recurrent network and utilize Riemannian operations to ensure that our method is faithful to the geometry of manifolds. The proposed method explores the distribution of the underlying data by minimizing the objective of updated parameters, and hence is capable of learning task-specific optimizations. We introduce a Riemannian implicit differentiation training scheme to achieve efficient training in terms of numerical stability and computational cost. Unlike conventional meta-optimization training schemes that need to differentiate through the whole optimization trajectory, our training scheme is only related to the final two optimization steps. In this way, our training scheme avoids the exploding gradient problem, and significantly reduces the computational load and memory footprint. We discuss experimental results across various constrained problems, including principal component analysis on Grassmann manifolds, face recognition, person re-identification, and texture image classification on Stiefel manifolds, clustering and similarity learning on symmetric positive definite manifolds, and few-shot learning on hyperbolic manifolds.

Characterizations of a spacetime admitting ψ-conformal curvature tensor

In this paper, we introduce ?-conformal curvature tensor, a new tensor that generalizes the conformal curvature tensor. At first, we deduce a few fundamental geometrical properties of ?-conformal curvature tensor and pseudo ?-conharmonically symmetric manifolds and produce some interesting outcomes. Moreover, we study ?-conformally flat perfect fluid spacetimes. As a consequence, we establish a number of significant theorems about Minkowski spacetime, GRW-spacetime, projective collineation. Moreover, we show that if a?-conformally flat spacetime admits a Ricci bi-conformal vector field, then it is either conformally flat or of Petrov type N. At last, we consider pseudo?conformally symmetric spacetime admitting harmonic ?-conformal curvature tensor and prove that the semi-symmetric energy momentum tensor and Ricci semi-symmetry are equivalent and also, the Ricci collineation and matter collineation are equivalent.

Some geometric and physical properties of pseudo m*-projective symmetric manifolds

In this study we introduce a new tensor in a semi-Riemannian manifold, named the M*-projective curvature tensor which generalizes the m-projective curvature tensor. We start by deducing some fundamental geometric properties of the M*-projective curvature tensor. After that, we study pseudo M*-projective symmetric manifolds (PM?S)n. A non-trivial example has been used to show the existence of such a manifold. We introduce a series of interesting conclusions. We establish, among other things, that if the scalar curvature ? is non-zero, the associated 1-form is closed for a (PM?S)n with divM* = 0. We also deal with pseudo M*-projective symmetric spacetimes, M*-projectively flat perfect fluid spacetimes, and M*-projectively flat viscous fluid spacetimes. As a result, we establish some significant theorems.

High-dimensional SO(4)-symmetric Rydberg manifolds for quantum simulation

We develop a toolbox for manipulating arrays of Rydberg atoms prepared in high-dimensional hydrogen-like manifolds in the regime of linear Stark and Zeeman effect. We exploit the SO(4) symmetry to characterize the action of static electric and magnetic fields as well as microwave and optical fields on the well-structured manifolds of states with principal quantum number n. This enables us to construct generalized large-spin Heisenberg models for which we develop state-preparation and readout schemes. Due to the available large internal Hilbert space, these models provide a natural framework for the quantum simulation of quantum field theories, which we illustrate for the case of the sine-Gordon and massive Schwinger models. Moreover, these high-dimensional manifolds also offer the opportunity to perform quantum information processing operations for qudit-based quantum computing, which we exemplify with an entangling gate and a state-transfer protocol for the states in the neighborhood of the circular Rydberg level.

Analytic torsion for arithmetic locally symmetric manifolds and approximation of L2-torsion

In this paper we define a regularized version of the analytic torsion for quotients of a symmetric space of non-positive curvature by arithmetic lattices. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat kernel. Then we study the limiting behavior of the analytic torsion as the lattices run through a sequence of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for sequences of principal congruence subgroups, which converge to 1 at a fixed finite set of places and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the L2-analytic torsion.

Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces and related Riemannian manifolds

We establish sharpened forms of the Hardy type identities and inequalities which are substantial improvements of Hardy inequalities for the operators −ΔH−(N−1)24 on the hyperbolic spaces HN. More precisely, we study the refined Hardy-Poincaré-Sobolev type inequalities in the spirit of Brezis-Vázquez [17] and Brezis-Marcus [15] on hyperbolic spaces, spherically symmetric Riemannian manifolds and more general Riemannian manifolds. Spherically symmetric manifolds are also of significant importance in physics and general relativity. Our approaches are to first establish the Hardy-Poincaré-Sobolev type identities using the notion of Bessel pairs on hyperbolic spaces and related Riemannian manifolds. Using a Hardy identity on upper half space, we also establish a sharp Sobolev inequality on a novel example of noncomplete and non-extendable Riemannian manifold with positive Ricci curvature. In particular, in dimension 3, our example shows that the completeness assumption of the manifolds in the rigidity result of Ledoux [47] cannot be removed.

Some Characterization Results on Generalized Weakly Symmetric Kenmotsu Manifolds

Some Characterization Results on Generalized Weakly Symmetric Kenmotsu Manifolds

Fourth Order Schrödinger Equation with Mixed Dispersion on Certain Cartan-Hadamard Manifolds

This paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on MN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^N$$\\end{document}, an N-dimensional Cartan-Hadamard manifold. Namely we consider 4NLSi∂tψ=-ΔM2ψ+βΔMψ+λ|ψ|2σψinR×M,ψ(0,·)=ψ0∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} i\\partial _t\\psi = -\\Delta _M^2\\psi +\\beta \\Delta _M \\psi +\\lambda |\\psi |^{2\\sigma }\\psi \\quad \ ext {in }\\mathbb {R}\ imes M,\\\\ \\psi (0,\\cdot )=\\psi _0\\in X, \\end{array}\\right. } \\end{aligned}$$\\end{document}where β≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\ge 0$$\\end{document}, λ={-1,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =\\{-1,1\\}$$\\end{document}, 0<σ<4/(N-4)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/(N-4)_+$$\\end{document}, ΔM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _M$$\\end{document} is the Laplace-Beltrami operator on M and X=L2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L^2 (M)$$\\end{document} or X=H2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=H^2 (M)$$\\end{document}. At first, we focus on the case where M is the hyperbolic space HN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^N$$\\end{document}. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to (4NLS) as well as scattering for small initial data provided that N≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 4$$\\end{document} and 0<σ<4/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/N$$\\end{document} if X=L2(HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L^2 (\\mathbb {H}^N)$$\\end{document} or 0<σ<4/(N-4)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/(N-4)_+$$\\end{document} if X=H2(HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=H^2 (\\mathbb {H}^N)$$\\end{document}. Next, we obtained weighted Strichartz estimates for radial solutions to (4NLS) on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.

Vector analysis on symmetric manifolds and Sobolev inequalities

Vector analysis on symmetric manifolds and Sobolev inequalities

Theta series and generalized special cycles on Hermitian locally symmetric manifolds

We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator representation and a construction which essentially comes from the thesis of Greg Anderson, we show that Poincar\'e duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermtian locally symmetric manifolds associated to $G$ to vector valued automorphic forms associated to the groups $G'=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{Sp}(m,m)$ which forms a reductive dual pair with $G$ in the sense of Howe.

Nonmetricity on Riemann–Cartan–Weyl manifold: Its physical and mathematical meaning and application

The physical meaning and mathematical meaning of nonmetricity on Weyl manifold and the application of dual affine connection for it are discussed. In viewpoint from affine connection, Weyl 1-form in Weyl manifold is recognized as nonmetricity and causes scale change with conformal invariance under parallel transport. On manifolds expressing spacetime and material space, Weyl 1-form is equivalent to the extra coupling field to field expressed as Riemannian manifold, protruding from Riemannian manifold. The scale change with conformal invariance corresponds to the similar change in the magnitude of the field with the volumetric distortion of each manifold. Weyl 1-form is the change rate of volume element (scale) in Riemannian manifold. Therefore, nonmetricity in Weyl manifold is defined as expansion or contraction rate as similar volumetric distortion due to the extra coupled field. The volumetric distortion as nonmetricity on Riemannian manifold can be canceled out by setting dual affine connection on Riemann–Cartan–Weyl manifold, which is used in statistical manifold. Moreover, the role of nonmetricity in modified gravity theory in the framework of symmetric teleparallel manifold and the meaning of nonmetricity on statistical manifold are discussed based on this concept.

Weakly symmetric pseudo–Riemannian nilmanifolds

In an earlier paper we developed the classification of weakly symmetric pseudo–Riemannian manifolds $G/H$, where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans-Lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo-Riemannian nilmanifolds $G/H$ from the classification for the case $G=N \rtimes H$ with $H$ compact and $N$ nilpotent. It turns out that there is a plethora of new examples that merit further study. Starting with that Riemannian case, we see just when a given involutive automorphism of $H$ extends to an involutive automorphism of $G$, and we show that any two such extensions result in isometric pseudo-Riemannian nilmanifolds. The results are tabulated in the last two sections of the paper.

On-shell correlators and color-kinematics duality in curved symmetric spacetimes

We define a perturbatively calculable quantity — the on-shell correlator — which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry. Afterwards, we compute tree-level on-shell correlators for biadjoint scalar (BAS) theory and the nonlinear sigma model (NLSM) and learn that color-kinematics duality is manifested at the level of fields under a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold. Last but not least, we present a field theoretic derivation of the fundamental BCJ relations for on-shell correlators following from the existence of certain conserved currents in BAS theory and the NLSM.

Riemannian submanifold framework for log-Euclidean metric learning on symmetric positive definite manifolds

This study presents a novel Riemannian submanifold (RS) framework for log-Euclidean metric learning on symmetric positive definite manifolds. Our method identifies the optimal RS without changing the original tangent space. The RS is spanned by multiple bases, and each data point is parameterized using these bases, such that the data can be represented more informatively compared to conventional approaches. In the RS, a new distance function is defined, and its derivative cannot be obtained trivially. We overcome this difficulty and provide a simple analytic form of the derivative. The proposed transformation matrix can take any form, which gives flexibility to our method and enables the creation of several variants. In experiments, our method and its variants surpass state-of-the-art metric learning methods in synthetic, material categorization, and action recognition problems.