Invariant Metrics Research Articles

The classical and quantum particle on a flag manifold

Abstract In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace–Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace–Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Given any quasi-countable, in particular, any countable inverse semigroup S , we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S . This allows us to unambiguously define the uniform Roe algebra of S , which we prove can be realized as a canonical crossed product of \ell^{\infty}(S) and S . We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0 . Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally \mathcal{L} -finite, and equivalently sparse as a metric space.

On a conjecture of Stolz in the toric case

In 1996 Stolz [Math. Ann. 304 (1996), pp. 785–800] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature.

Characterization of Holomorphic Invariant Complex Finsler Metrics and Schwarz Lemma on the Polyball

Characterization of Holomorphic Invariant Complex Finsler Metrics and Schwarz Lemma on the Polyball

A harmonic property of right invariant priors

AbstractIt is shown that, on any Lie group, the density ratio of the right invariant measure to the left invariant measure is harmonic with respect to the left invariant Riemannian metric. This result is applied to the Bayesian prediction theory on group invariant statistical models. A method of constructing Bayesian prior distributions that asymptotically dominate the right invariant priors is provided.

Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Let P:=U(p+q)/U(p)×U(q) be the complex Grassmann manifold and F:T1,0P→[0,+∞) be an arbitrary U(p+q)-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U(p+q)-invariant Kähler metric on P. In particular on the complex projective space CPn=U(n+1)/U(n)×U(1), there exists no U(n+1)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U(p+q)-invariant Kähler metrics on P, nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases.

Partial actions on quotient spaces and globalization

Given a partial action of a topological group G on a space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable, and having invariant metric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space of X; applications to invariant metrics and inverse limits are presented.

Covariant Derivatives on Homogeneous Spaces: Horizontal Lifts and Parallel Transport

We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on the Lie group. This point of view allows for a characterization of parallel vector fields along curves. Moreover, metric invariant covariant derivatives on a reductive homogeneous space equipped with an invariant pseudo-Riemannian metric are characterized. As a by-product, a new proof for the existence of invariant covariant derivatives on reductive homogeneous spaces and their the one-to-one correspondence to certain bilinear maps is obtained.

Geodesics on adjoint orbits of SL(n, ℝ)

In this paper, we examine the geodesics on adjoint orbits of [Formula: see text] that are equipped with [Formula: see text]-invariant metrics, where [Formula: see text] is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain [Formula: see text]-flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for [Formula: see text].

Impurity-doped stable domain walls in spherically symmetric spacetimes

In this work, radially symmetric domain wall solutions in the presence of impurities are investigated for both flat and curved D+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D+1$$\\end{document} spacetimes, with geometry generated by a rotationally invariant background metric. We have examined the constraints placed upon the model by Derrick’s theorem, and found out the Bogomol’nyi bound and equations of the symmetric restriction to this theory. Impurity-doped versions of a ϕ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^4$$\\end{document} model in two dimensions and a model with logarithmic potential in a Schwarzschild background have been explicitly worked out. The resulting configurations have been compared with those found in the homogeneous version of the theory, so that the effect of impurities in the form of solutions may be better appreciated. We have also generalized to higher dimensions some of the results that had been presented in the recent literature. These results relate to the possibility of BPS-preserving impurities, which we have found to still exist in the spacetimes considered in this work. We also investigate ways in which these results may be extended in a curved background.

Conformally Invariant Metrics and Lack of Hölder Continuity

The modulus metric between two points in a subdomain of Rn,n≥2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n, n\\ge 2,$$\\end{document} is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolic-type metrics that have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

We study the relation between two special classes of Riemannian Lie groups [Formula: see text] with a left-invariant metric [Formula: see text]: The Einstein Lie groups, defined by the condition [Formula: see text], and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups [Formula: see text] are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the [Formula: see text]-invariant geodesic orbit metrics on Lie groups [Formula: see text] for a wide class of subgroups [Formula: see text] that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

Sobolev metrics on spaces of manifold valued curves

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

Lee metrics on groups

In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and not to have interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $ G $ is a torsion-free group or a finite group of odd order, we prove that $ G $ has a Lee metric if and only if $ G $ is cyclic. Also, if $ G $ is a group admitting Lee metrics then $ G \times \mathbb{Z}_2^k $ always has Lee metrics for every $ k \in \mathbb{N} $. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $ \le 31 $ indicating which of them have (or have not) Lee metrics and why (not).

Non Naturally Reductive Einstein Metrics on the Orthogonal Group via Real Flag Manifolds

We obtain new invariant Einstein metrics on the compact Lie groups SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document} which are not naturally reductive. This is achieved by using the real flag manifolds SO(k1+⋯+kp)/SO(k1)×⋯×SO(kp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(k_1+\\cdots +k_p)/{{\\,\ extrm{SO}\\,}}(k_1)\ imes \\cdots \ imes {{\\,\ extrm{SO}\\,}}(k_p)$$\\end{document} and by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document}.

Lie group geometry. Invariant metrics and dynamical systems, dual algebra, and their applications in the group analysis of a one-dimensional kinetic equation

Lie group geometry. Invariant metrics and dynamical systems, dual algebra, and their applications in the group analysis of a one-dimensional kinetic equation

Metrics related to oscillator algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

Identifying and profiling structural similarities between Spike of SARS-CoV-2 and other viral or host proteins with Machaon

Using protein structure to predict function, interactions, and evolutionary history is still an open challenge, with existing approaches relying extensively on protein homology and families. Here, we present Machaon, a data-driven method combining orientation invariant metrics on phi-psi angles, inter-residue contacts and surface complexity. It can be readily applied on whole structures or segments—such as domains and binding sites. Machaon was applied on SARS-CoV-2 Spike monomers of native, Delta and Omicron variants and identified correlations with a wide range of viral proteins from close to distant taxonomy ranks, as well as host proteins, such as ACE2 receptor. Machaon’s meta-analysis of the results highlights structural, chemical and transcriptional similarities between the Spike monomer and human proteins, indicating a multi-level viral mimicry. This extended analysis also revealed relationships of the Spike protein with biological processes such as ubiquitination and angiogenesis and highlighted different patterns in virus attachment among the studied variants. Available at: https://machaonweb.com.

Invariant metrics on current Lie algebras

In this work we state conditions for a current Lie algebra to admit an invariant metric, where is a quadratic Lie algebra and is an associative and commutative algebra with unit. We show that if is an indecomposable quadratic Lie algebra, then admits an invariant metric if and only if also admits an invariant metric. In addition, we prove a theorem similar to the double extension for , where is an indecomposable, nilpotent and quadratic Lie algebra.

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.