Compact Homogeneous Spaces Research Articles

Single radius spherical cap discrepancy on compact two-point homogeneous spaces

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a d-dimensional manifold M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} endowed with a distance ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} so that (M,ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathcal {M}, \\rho )$$\\end{document} is a two-point homogeneous space and with the Riemannian measure μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}, we provide conditions on r such that if Dr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_r$$\\end{document} denotes the discrepancy of the ball of radius r, then, for an absolute constant C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C>0$$\\end{document} and for every set of points {xj}j=1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x_j\\}_{j=1}^N$$\\end{document}, one has ∫M|Dr(x)|2dμ(x)⩾CN-1-1d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{\\mathcal {M}} |D_{r}(x)|^2\\, d\\mu (x)\\geqslant C N^{-1-\\frac{1}{d}}$$\\end{document}. The conditions on r that we have depend on the dimension d of the manifold and cannot be achieved when d≡1(mod4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\equiv 1 \\ ( {\ ext {mod}}4)$$\\end{document}. Nonetheless, we prove a weaker estimate for such dimensions as well.

Correction: The geometry of compact homogeneous spaces with two isotropy summands

Correction: The geometry of compact homogeneous spaces with two isotropy summands

Equigeodesic vectors on compact homogeneous spaces with equivalent isotropy summands

Equigeodesic vectors on compact homogeneous spaces with equivalent isotropy summands

Strictly positive definite non-isotropic kernels on two-point homogeneous manifolds: the asymptotic approach

We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

We classify all the 6-dimensional unimodular Lie algebras g admitting a complex structure with non-zero closed (3,0)-form. This gives rise to 6-dimensional compact homogeneous spaces M=Γ﹨G, where Γ is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection ∇ε,ρ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being Hermitian-Yang-Mills, we prove that if a compact non-Kähler homogeneous space M=Γ﹨G admits an invariant solution with respect to some non-flat connection ∇ in the family ∇ε,ρ, then M is a nilmanifold with underlying Lie algebra h3, a solvmanifold with underlying algebra g7, or a quotient of the semisimple group SL(2, C). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.

Point pattern analysis and classification on compact two-point homogeneous spaces evolving time.

This paper introduces a new modeling framework for the statistical analysis of point patterns on a manifold defined by a connected and compact two-point homogeneous space, including the special case of the sphere. The presented approach is based on temporal Cox processes driven by a -valued log-intensity. Different aggregation schemes on the manifold of the spatiotemporal point-referenced data are implemented in terms of the time-varying discrete Jacobi polynomial transform of the log-risk process. The n-dimensional microscale point pattern evolution in time at different manifold spatial scales is then characterized from such a transform. The simulation study undertaken illustrates the construction of spherical point process models displaying aggregation at low Legendre polynomial transform frequencies (large scale), while regularity is observed at high frequencies (small scale). K-function analysis supports these results under temporal short, intermediate and long range dependence of the log-risk process.

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

This paper is concerned with sample path properties of real-valued isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field and then exploit this result to establish an exact uniform modulus of continuity for its sample paths.

On the fibre structure of compact homogeneous spaces

The paper considers properties of several bundles (natural, structural and Borel ones) related to compact homogeneous spaces. Some results on elements of these bundles are proved, illustrative examples are given, counterexamples to some naturally arising assumptions are put forward.

On the fibre structure of compact homogeneous spaces

В статье рассматриваются свойства нескольких расслоений, связанных с компактными однородными пространствами, - натурального, структурного и борелевского. Доказаны некоторые утверждения об элементах этих расслоений, приведены иллюстрирующие их примеры, а также и контрпримеры к некоторым естественно возникающим предположениям. Библиография: 29 наименований.

Infinite families of homogeneous Bismut Ricci flat manifolds

Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order [Formula: see text] and (up to coverings) they can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.

Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov–Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.

On Randers geodesic orbit spaces

In this paper, we consider invariant Randers g.o. metrics on compact homogeneous spaces. In particular, we obtain a concrete characterization of the corresponding navigation datas of these metrics. As an application, we discuss all invariant Randers g.o. metrics on compact homogeneous spaces with two isotropy summands and compact homogeneous spaces with simple isotropy subgroups, which provide a large number of new examples of Finsler g.o. spaces.

On the stability of homogeneous Einstein manifolds II

For any G $G$ -invariant metric on a compact homogeneous space M = G / K $M=G/K$ , we give a formula for the Lichnerowicz Laplacian restricted to the space of all G $G$ -invariant symmetric 2-tensors in terms of the structural constants of G / K $G/K$ . As an application, we compute the G $G$ -invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with b 2 ( M ) = 1 $b_2(M)=1$ . This allows to deduce the G $G$ -stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension$5$is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

Series expansions among weighted Lebesgue function spaces and applications to positive definite functions on compact two-point homogeneous spaces

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

Stochastic comparison for elliptically contoured random fields

This paper presents necessary and sufficient conditions for the peakedness comparison and convex ordering between two elliptically contoured random fields about their centers. A somewhat surprising finding is that the peakedness comparison for the infinite dimensional case differs from the finite dimensional case. For example, a Student’s t distribution is known to be more heavy-tailed than a normal distribution, but a Student’s t random field and a Gaussian random field are not comparable in terms of the peakedness. In particular, the peakedness comparison and convex ordering are made for isotropic elliptically contoured random fields on compact two-point homogeneous spaces.

DIAMETER AND LAPLACE EIGENVALUE ESTIMATES FOR COMPACT HOMOGENEOUS RIEMANNIAN MANIFOLDS

Let G be a compact connected Lie group and let K be a closed subgroup of G. In this paper, we study whether the functional 𝔤 ↦ ⋋1 (G/K, 𝔤) diam (G/K, 𝔤)2 is bounded among G-invariant metrics 𝔤 on G/K. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when K is trivial; the only particular cases known so far are when G is abelian, SU(2), and SO(3). In this article, we prove the existence of the mentioned upper bound for every compact homogeneous space G/K having multiplicity-free isotropy representation.

Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes \(S_r(L^2(G/H))\) for all \(0<r \le \infty .\) We go on to provide a symbolic characterization for r-nuclear, \(0< r \le 1,\) pseudo-differential operators on \(L^{p}(G/H)\) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space \(G/H \times \widehat{G/H}.\) Finally, we present an application of aforementioned results in the context of the heat kernels.

On Weakly Reflective Submanifolds in Compact Isotropy Irreducible Riemannian Homogeneous Spaces

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

Aperiodic order and spherical diffraction, II: translation bounded measures on homogeneous spaces

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on mathbb {R}^n. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.