Symmetric Spaces Research Articles

Buildings, valuated matroids, and tropical linear spaces

AbstractAffine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.

New gravastar model in generalised cylindrically symmetric space–time and prediction of mass limit

We present a class of new Gravastar solutions following the works of Mazur and Mottola for a Gravitational Bose–Einstein Condensate (GBEC) star in generalised cylindrically symmetric space–time. A stable gravastar consists of 3 distinct regions, namely: (i) an interior de-Sitter space (p=−ρ), which exerts an outwards repulsive force at all points on the thin shell, (ii) an intermediate thin shell with a slice of finite length separating the interior and exterior regions is supposed to be consisting of an ultra-relativistic stiff fluid, with the equation of state p=ρ and (iii) an exterior vacuum region. This thin shell, which is considered as the critical surface for the quantum phase transition, replaces both the classical de-Sitter space and Schwarzschild event horizon. The new solutions are free from any singularities. The energy density, total energy, proper length, and the entropy of this shell are explored in this model. From the thin shell solution and using Lanczos equations for stress energy density, we have predicted the mass contained into the gravastar shell. The stability of the gravastar model is analysed through the consideration of gravitational surface redshift and entropy calculation. We have also obtained a constraint on the possible mass of the thin shell without violating the condition for stable gravastar configuration. All these features indicate that the present model is physically viable.

Group Actions and Harmonic Analysis in Number Theory

This workshop focuses on new problems and new methods at the interface of harmonic analysis (taken in a very broad sense) and ergodic theory, with applications focused on number theory. Special emphasis is put on equidistribution problems on arithmetic symmetric spaces, effective methods in homogeneous dynamics, periods of automorphic forms, families of L -functions over number fields and function fields, and applications of Fourier uniqueness.

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Nicolai maps with four-fermion interactions

Nicolai maps offer an alternative description of supersymmetric theories via nonlinear and nonlocal transformations characterized by the so-called ‘free-action’ and ‘determinant-matching’ conditions. The latter expresses the equality of the Jacobian determinant of the transformation with the one obtained by integrating out the fermions, which so far have been considered only to quadratic terms. We argue that such a restriction is not substantial, as Nicolai maps can be constructed for arbitrary nonlinear sigma models, which feature four-fermion interactions. The fermionic effective one-loop action then gets generalized to higher loops and the perturbative tree expansion of such Nicolai maps receives quantum corrections in the form of fermion loop decorations. The ‘free-action condition’ continues to hold for the classical map, but the ‘determinant-matching condition’ is extended to an infinite hierarchy in fermion loop order. After general considerations for sigma models in four dimensions, we specialize to the case of ℂPN symmetric spaces and construct the associated Nicolai map. These sigma models admit a formulation with only quadratic fermions via an auxiliary vector field, which does not simplify our analysis.

Equivariant formality of the isotropy action on (ℤ2 ⊕ ℤ2)-symmetric spaces

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalization, it is even known that their isotropy action is equivariantly formal. In this paper, we show that [Formula: see text]-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field (cf. [66]) would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as “killing the genus by a purely transcendental extension.” Yet another application deals with the investigation of “eigenvalue rigidity” of Zariski-dense subgroups (cf. [64]), which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the Finiteness Conjecture and the genus problem for simple algebraic groups of type F4.

On local intertwining periods

We prove the absolute convergence, functional equations and meromorphic continuation of local intertwining periods on parabolically induced representations of finite length for certain symmetric spaces over local fields of characteristic zero, including Galois pairs as well as pairs of Prasad and Takloo-Bighash type. Furthermore, for a general symmetric space we prove a sufficient condition for distinction of an induced representation in terms of distinction of its inducing data. Both results generalize previous results of the first two named authors. In particular, for both we remove a boundedness assumption on the inducing data and for the second we further remove any assumption on the symmetric space. Moreover, we extend the field of coefficients from p-adic to any local field of characteristic zero. In the case of p-adic symmetric spaces, combined with the necessary conditions for distinction that follow from the geometric lemma, this provides a necessary and sufficient condition for distinction of representations induced from cuspidal.

CAT(0) spaces of higher rank II

This belongs to a series of papers motivated by Ballmann’s Higher Rank Rigidity Conjecture. We prove the following. Let X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} be a CAT(0) space with a geometric group action Γ↷X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma \\curvearrowright X$\\end{document}. Suppose that every geodesic in X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} lies in an n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-flat, n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n\\geq 2$\\end{document}. If X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} contains a periodic n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-flat which does not bound a flat (n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(n+1)$\\end{document}-half-space, then X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.

A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation

As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion and superdiffusion. Due to the fact that fractional operators act on both temporal and spatial derivative terms in TSFDE, efficient solving for TSFDE is important, where the key is solving the corresponding discrete system efficiently. In this paper, we derive a Galerkin–Legendre spectral all-at-once system from the TSFDE, and then we develop a preconditioner to solve this system. Symmetry property of the coefficient matrix in this all-at-once system is destroyed so that the deduced all-at-once system is more convenient for parallel computing than the traditional timing-step scheme, and the proposed preconditioner can efficiently solve the corresponding all-at-once system from TSFDE with nonsmooth solution. Moreover, some relevant theoretical analyses are provided, and several numerical results are presented to show competitiveness of the proposed method.

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Fixed Point Theory in Extended Parametric Sb-Metric Spaces and Its Applications

This article introduces the novel concept of an extended parametric Sb-metric space, which is a generalization of both Sb-metric spaces and parametric Sb-metric spaces. Within this extended framework, we first establish an analog version of the Banach fixed-point theorem for self-maps. We then prove an improved version of the Banach contraction principle for symmetric extended parametric Sb-metric spaces, using an auxiliary function to establish the desired result. Finally, we provide illustrative examples and an application for determining solutions to Fredholm integral equations, demonstrating the practical implications of our work.

Approaching the isoperimetric problem in HCm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{{\\mathbb {C}}}^m$$\\end{document} via the hyperbolic log-convex density conjecture

We prove that geodesic balls centered at some base point are uniquely isoperimetric sets in the real hyperbolic space HRn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{{\\mathbb {R}}}^n$$\\end{document} endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are uniquely isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.

On the physical viability of black hole solutions in Einsteinian Cubic Gravity and its generalisations

In this note, we discuss the pathological nature of black holes in Einsteinian Cubic gravity and its extensions. We compute the equations for the odd perturbations and show how spherically symmetric solutions that asymptotically approach a maximally symmetric space (Minkowski, de Sitter or anti-de Sitter) are associated to having an asymptotically degenerate principal part of the equations. We use these results to argue that the encountered problems will be generic for any other cubic or higher-order with a reduced linear spectrum around maximally symmetric spaces except the well-known healthy case of f(R). We highlight that these pathologies are only alarming when the theory is regarded as a complete theory, but not when considered as a perturbative correction to GR (as in e.g. the effective field theory framework) since the low-energy physics remains safe from them. Our results thus cast doubts on possible resolutions of the singularities or non-perturbative effects on horizons based on these theories.

VSTNet: Robust watermarking scheme based on voxel space transformation for diffusion tensor imaging images

Deep watermarking has been successfully applied to real-value domain (e.g., CT, MRI, DWI) medical images for copyright protection; however, manifold-value domain images, such as diffusion tensor imaging (DTI), have received less academic attention. The non-Euclidean nature of DTI images hinders deep models from generating plausible DTI images because it is difficult to guarantee that each generated DTI image voxel lies on a 3 × 3 symmetric positive definite manifold. In this paper, we propose a robust watermarking scheme based on voxel space transformation for DTI images. First, a voxel space transformation is proposed, which effectively solves the problem of fiber orientation estimation in Riemann networks for DTI images and obtains the eigenvalues and eigenvectors of each voxel using singular value decomposition (SVD) to transform the DTI voxels from the 3 × 3 symmetric positive definite manifold space to Euclidean space. Second, SVD is combined with a deep neural network to extract the high-level features of each voxel eigenvalue in the DTI images and embedding watermarking messages, avoiding the false positive errors problem in traditional watermarking algorithms based on an SVD transform. Finally, using a deep neural network combining multiscale dilated convolution, dense residual connection and channel attention, the algorithm demonstrates high robustness against various intentional or unintentional attacks on DTI images while ensuring the good visual quality and diffusion characteristics of the DTI images embedded with watermarking messages. Experiments show that the watermarking message extraction accuracy reaches 97.8% for Crop (p = 0.7) attacks and 95.4% for Gaussian Blur (k = 7) attacks.

Quantum (in)stability of maximally symmetric space-times

Classical gravity coupled to a CFT4 (matter) is considered. The effect of the quantum dynamics of matter on gravity is studied around maximally symmetric spaces (flat, de Sitter and Anti de Sitter). The structure of the graviton propagator is modified and non-trivial poles appear due to matter quantum effects. The position and residues of such poles are mapped as a function of the relevant parameters, the central charge of the CFT4, the two R2 couplings of gravity as well as the curvature of the background space-time. The instabilities induced are determined. Such instabilities can be important in cosmology as they trigger the departure from de Sitter space and in some regions of parameters are more important than the well-known scalar instabilities. It is also determined when the presence of such instabilities is unreliable if the associated scales are larger than the “species” cutoff of the gravitational theory.

Revisiting linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Revisiting linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Manifolds with harmonic Weyl tensor and nonnegative curvature operator of the second kind

We show that a compact manifold with harmonic Weyl tensor and nonnegative curvature operator of the second kind is globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. Moreover, we obtain that a compact manifold with harmonic curvature and nonnegative curvature operator of the second kind is isometric to a space of constant curvature. These improve Kashiwada's result that Riemannian manifolds with harmonic curvature tensor for which the curvature operator of the second kind is nonnegative and positive are locally symmetric spaces and constant curvature spaces, respectively.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over R, C or H. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

Rolling Stiefel Manifolds Equipped with α-Metrics

We discuss the rolling, without slipping and without twisting, of Stiefel manifolds equipped with α-metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely, by investigating the intrinsic rolling of normal naturally reductive homogeneous spaces. This gives evidence as to why a seemingly straightforward generalization of the intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous spaces is not possible, in general. For a given control curve, we derive a system of explicit time-variant ODEs whose solutions describe the desired rolling. These findings are applied to obtain the intrinsic rolling of Stiefel manifolds, which is then extended to an extrinsic one. Moreover, explicit solutions of the kinematic equations are obtained, provided that the development curve is the projection of a not necessarily horizontal one-parameter subgroup. In addition, our results are put into perspective with examples of the rolling Stiefel manifolds known from the literature.