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We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces $ G/\Gamma $ G / Γ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks. [ABSTRACT FROM AUTHOR]

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles. [ABSTRACT FROM AUTHOR]

In this article, we study the geodesic orbit Randers spaces of the form ( G / H , F ) {(G/H,F)} , such that G is one of the compact classical Lie groups SO ⁢ ( n ) {{\mathrm{S}}{\mathrm{O}}(n)} , SU ⁢ ( n ) {{\mathrm{S}}{\mathrm{U}}(n)} , Sp ⁢ ( n ) {{\mathrm{S}}{\mathrm{p}}(n)} , and H is a diagonally embedded product H 1 × ⋯ × H s {H_{1}\times\cdots\times H_{s}} , where H i {H_{i}} is of the same type as G. Such spaces include spheres, Stiefel manifolds, Grassmann manifolds, and flag manifolds. The present work is a contribution to the study of geodesic orbit Randers spaces ( G / H , F ) {(G/H,F)} with H semisimple. We construct new examples of non-Riemannian Randers g.o. metrics in homogeneous bundles over generalized Stiefel manifolds which are not naturally reductive. Also, we obtain the specific expressions of these Randers g.o. metrics. [ABSTRACT FROM AUTHOR]

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we give the flag curvature formula of general (α, β)-metrics of Berwald type. We study conformally related (α, β)-metrics, especially general (α, β)-metrics that are conformally related to invariant (α, β)-metrics. Also, a necessary and sufficient condition for a Finsler metric conformally related to an (α, β)-metric is given and conformally related Douglas Randers metrics are studied. Finally, we present some examples of conformally related (α, β)-metrics. [ABSTRACT FROM AUTHOR]

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 4 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. [ABSTRACT FROM AUTHOR]

The paper studies the energy-momentum tensor components for admissible electromagnetic fields in a nonholonomic system given by the group operation in homogeneous spaces. Compact expressions are obtained for Maxwell field equations. [ABSTRACT FROM AUTHOR]

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$ , a $G$ -torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the projective action of the Lie group SL(3;R) on the homogeneous space RP2. In particular, we have studied the action of the subgroups of SL(3;R) on the non-degenerate conics in the space RP2. Using the Iwasawa decomposition of SL(2;R), we demonstrate that the isotropy subgroup of the projective unit circle is isomorphic to PSL(2;R) under certain conditions. [ABSTRACT FROM AUTHOR]

This paper has features of mixed nature. It contains new results, in particular, on semitopological groups. We also pose some problems, introduce new definitions and describe in details certain techniques we need below, providing the proofs of not so well-known theorems for the sake of completeness. Hence, this article can be treated also as a kind of a short survey. [ABSTRACT FROM AUTHOR]

Kathryn F. Porter wrote a nice paper about several definitions of local homogeneity [Local homogeneity, JP Journal of Geometry and Topology 9 (2009), 129–136]. In this paper, she mentions that G. S. Ungar defined a uniformly locally homogeneous space [Local homogeneity, Duke Math. J. 34 (1967), 693–700]. We realized that this notion is very similar to what we call the uniform property of Effros [On Jones' set function 풯 and the property of Kelley for Hausdorff continua, Topology Appl. 226 (2017), 51–65]. Here, we compare the uniform property of Effros with the uniform local homogeneity. We also consider other definitions of local homogeneity given in Porter's paper and compare them with the uniform property of Effros. We show that in the presence of compactness, the uniform property of Effros is equivalent to uniform local homogeneity and the local homogeneity according to Ho. With this result, we can change the hypothesis of the uniform property of Effros in Jones' and Prajs' decomposition theorems to uniform local homogeneity and local homogeneity according to Ho. We add to these two results the fact that the corresponding quotient space also has the uniform property of Effros. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss an important discourse, as most literatures highlighted the dimension of partial flag in a certain manner, according to a fixed definition that does not include all possible cases. In the present paper the existence of two different cases of flags in the partial flag varieties of the groups SO(n,c) and SP(2n,c) is explained wholly. In addition to this we will proceed by calculating the dimensions for all the given cases after pertaining the signature of these flags that are studied. [ABSTRACT FROM AUTHOR]

Some curvature functionals which are defined according to the quadratic curvature invariants were studied on a special class of space-times. We exactly determine metrics that are critical for those considering curvature functionals, through homogeneous classes. [ABSTRACT FROM AUTHOR]

The curvature characteristics of particular classes of Finsler spaces, such as homogeneous Finsler spaces, are one of the major issues in Finsler geometry. In this paper, we have obtained the expression for S-curvature in homogeneous Finsler space with a generalized Matsumoto metric and demonstrated that the homogeneous generalized Matsumoto space with isotropic S-curvature has to vanish the S-curvature. We have also derived the expression for the mean Berwald curvature by using the formula of S-curvature. [ABSTRACT FROM AUTHOR]

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic. [ABSTRACT FROM AUTHOR]

We investigate the action of the Lie group SL(3, R) on the two-dimensional homogeneous space. All the one-parameter subgroups (up to conjugacy) of SL(3, R) are considered. We discuss the orbits and curvatures of these one-parameter subgroups. We also classify these subgroups in terms of Fixed points. [ABSTRACT FROM AUTHOR]

We show that a four-dimensional homogeneous manifold with half-harmonic Weyl curvature tensor is symmetric or homothetic either to the only nonsymmetric anti-self-dual homogeneous manifold or to the 3-symmetric space. [ABSTRACT FROM AUTHOR]

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 E 3 (κ , τ) 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 E 3 (κ , τ) , becoming equalities if and only if the surface is a Hopf torus in a Berger sphere. [ABSTRACT FROM AUTHOR]

In this paper, we consider homogeneous spaces of functions defined on the whole real axis with values in a complex Banach space. A new class of functions from a homogeneous space that are almost uniformly periodic at infinity is introduced and examined. Four definitions of such functions are proposed and their equivalence is proved. Fourier series of functions almost periodic at infinity are constructed and their properties are analyzed. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules. [ABSTRACT FROM AUTHOR]

This paper presents an operator theory approach for the abstract structure of Banach function modules over coset spaces of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Let μ be the normalized G-invariant measure over the homogeneous space G / H associated to the Weil's formula and 1 ≤ p < ∞ . We then introduce the notion of convolution left-module action of L 1 (G / H , μ) on the Banach function spaces L p (G / H , μ) . [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of slowly varying and almost periodic at infinity distributions from harmonic spaces; a number of spaces of homogeneous functions are considered. The notion of a harmonic space of distributions is introduced; this space is constructed by a homogeneous functional spaces. Properties of harmonic spaces of distributions endowed with the structure of Banach modules are studied. We prove that such spaces are isometrically isomorphic to the corresponding homogeneous functional spaces. Based on the definitions of slowly varying and almost periodic at infinity functions from a homogeneous space, we introduce the notions of slowly varying and almost periodic at infinity distributions from a harmonic space. Using methods of abstract harmonic analysis, we construct Fourier series of almost periodic distributions at infinity and examine their properties. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules. [ABSTRACT FROM AUTHOR]

In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, H ⊂ K ⊂ G , such that G / K is a compact, connected, irreducible, symmetric space, and the isotropy representation of G / H has exactly two inequivalent, irreducible summands. We prove that the left metric 〈 · , · 〉 t 1 , t 2 on G defined by the first equation, must be an A -metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant B -metrics, such as 〈 · , · 〉 t 1 , t 2 . [ABSTRACT FROM AUTHOR]

Let X be a real algebraic variety (resp. nonsingular real algebraic variety) and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C ∞) map f : X → Y can be approximated by regular maps in the C 0 (resp. C ∞) topology if and only if it is homotopic to a regular map. Taking Y = S p , the unit p -dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C ∞ map from S n into S n can be approximated by regular maps in the C ∞ topology. Up to now such a result has only been known for five special values of n , namely, n = 1 , 2 , 3 , 4 or 7. [ABSTRACT FROM AUTHOR]

In this paper we prove the following weighted Sobolev inequality in a bounded domain Ω⊂Rn, n≥1, of a homogeneous space (Rn, ρ, wdx), under suitable compatibility conditions on the positive weight functions (v,w,ω1,ω2,...,ωn) and on the quasi-metric ρ, (∫Ω|f|qvwdz)1/q≤C∑i=1N(∫Ω|fzi|pωiMSwdz)1/p,f∈Lip0(Ω), where q≥p>1 and MS denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived. [ABSTRACT FROM AUTHOR]

We classify homogeneous four‐dimensional manifolds satisfying a curvature condition which naturally generalizes the Einstein one. Our work leads to new examples of self‐similar solutions of the two‐loop renormalization group flow. [ABSTRACT FROM AUTHOR]

We introduce a category of fuzzy convergence groups, FCONVGRP a subcategory of the category of fuzzy convergence spaces, FCONV. Viewing I as a complete Heyting algebra, we prove that the category of I -tolerance groups, I -TOLGRP is isomorphic to a subcategory of FCONVGRP. Since FCONV is a topological universe, and thereby possesses function space structure, upon invoking this, we are able, among others, to show that FCONVGRP is topological, and more importantly, it enables us to obtain a compatible fuzzy convergence function space structure on group of homeomorphisms. It is noticeable, however, that the category of fuzzy neighborhood convergence groups, FNCONVGRP — a supercategory of the well-known category FNS, of fuzzy neighborhood spaces, as well as the category of fuzzy neighborhood groups, FNGRP — a subcategory of FNCONVGRP exhibit nice relationships with FCONVGRP. It is important to note that the objects of FCONVGRP are homogeneous, this paves the way to present two pertinent characterization theorems on fuzzy convergence groups. Finally, introducing a category PSTOPGRP, of pseudotopological groups, we reveal the embeddings of FTOPGRP and PSTOPGRP into FCONVGRP. [ABSTRACT FROM AUTHOR]

We introduce almost diagonal matrices in the setting of (anisotropic) discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation spaces, and it is shown that the class of almost diagonal matrices is closed under matrix multiplication. We then connect the results to the continuous setting and show that the "change of frame" matrix for a pair of time-frequency frames, with suitable decay properties, is almost diagonal. As an application of this result, we consider a construction of compactly supported frame expansions for homogeneous decomposition spaces of Triebel-Lizorkin type and for the associated modulation spaces. [ABSTRACT FROM AUTHOR]

We prove that irreducible symmetric domains are uniquely determined by the homotopy equivalence classes of their Shilov boundaries. [ABSTRACT FROM AUTHOR]

Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data. [ABSTRACT FROM AUTHOR]

We obtain a necessary and sufficient condition for the existence of equivariant real structures on complex symmetric spaces for semisimple algebraic groups and discuss how to determine the number of equivalence classes for such structures. [ABSTRACT FROM AUTHOR]

Let H be a compact subgroup of a locally compact group G. We first investigate some (operator) (co)homological properties of the Fourier algebra A(G/H) of the homogeneous space G/H such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that A(G/H) is operator approximately biprojective if and only if G/H is discrete. We also show that A(G/H)** is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on A(G/H). [ABSTRACT FROM AUTHOR]

We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant ψ ¯ c (X) , we show that | X | ≤ π χ (X) c (X) ψ ¯ c (X) for any topological space X. If X is Hausdorff then ψ ¯ c (X) ≤ ψ c (X) ; this gives a strengthening of a theorem of Shu-Hao [24]. We also prove that | X | ≤ 2 p w L c (X) t (X) p c t (X) for a homogeneous Hausdorff space X. The invariant p w L c (X) , introduced in [9], is bounded above by both L(X) and c(X). Our result thus improves the bound | X | ≤ 2 L (X) t (X) p c t (X) for homogeneous Hausdorff spaces X [13] and represents a new extension of de la Vega's Theorem [15] into the Hausdorff setting. Moreover, we show p w L (X) ≤ a L (X) , demonstrating that 2 p w L (X) χ (X) is not a cardinality bound for all Hausdorff spaces. This answers a question of Bella and Spadaro [9]. A further theorem on covers by G κ c -sets lead to cardinality bounds involving the linear Lindelöf degree l L (X) , a weakening of L(X). It was shown in [5] that | X | ≤ 2 l L (X) F (X) ψ (X) for Tychonoff spaces. We show the consistency of a) | X | ≤ 2 l L (X) F (X) ψ c (X) if X is Hausdorff, and b) | X | ≤ 2 l L (X) F (X) p c t (X) if X is Hausdorff and homogeneous. If X is additionally regular, the former consistently improves the result from [5]. The latter gives a consistent improvement of the inequality | X | ≤ 2 L (X) t (X) p c t (X) for homogeneous Hausdorff spaces. [ABSTRACT FROM AUTHOR]

We construct a space of quantum states and an algebra of quantum observables, over the set of all metrics of arbitrary but fixed signature, defined on a manifold. The construction is diffeomorphism invariant, and unique up to natural isomorphisms. [ABSTRACT FROM AUTHOR]

Properties of homogeneous spaces having an invariant -structure (a generalization of the structure of a homogeneous dual manifold) are studied. Simply connected homogeneous spaces of dimension with such a structure are described up to a diffeomorphism. [ABSTRACT FROM AUTHOR]

We consider a shape optimization problem for the first mixed Steklov–Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem. [ABSTRACT FROM AUTHOR]

Given a homogeneous pseudo-Riemannian space (G / H , ⟨ , ⟩) , a geodesic γ : I → G / H is said to be two-step homogeneous if it admits a parametrization t = ϕ (s) (s affine parameter) and vectors X, Y in the Lie algebra g , such that γ (t) = exp (t X) exp (t Y) · o , for all t ∈ ϕ (I) . As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics ⟨ , ⟩ on the unimodular Lie group S L (2 , R) such that (S L (2 , R) , ⟨ , ⟩) is a two-step g.o. space. [ABSTRACT FROM AUTHOR]

In this paper, we show that the projection of every compact Riemannian manifold of positive curvature onto a rank one symmetric space is harmonic. As a corollary, an in finite family of distinct harmonic morphisms with minimal circle Fibers from the 7-dimensional homogeneous Aloff-Wallach spaces of positive curvature onto the 6-dimensional ag manifolds is given. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group and H be a closed subgroup of G. It is well-known that G/H as a homogeneous space aAdbmsittsraa csttr.ongly quasi invariant measure and the linear mapping TH of L1(G) into L1(G/H) is bounded and surjective. In this note it is shown that by means of complex interpolation theorem, that under restrictions on weight function w, the mapping TH of weighted spaces Lp (G, w) into Lp(G/H, w) is well-defined, bounded linear and surjective, for 1 < p < oe. [ABSTRACT FROM AUTHOR]

We study invariant Einstein metrics on the Stiefel manifold V k R n ≅ SO (n) / SO (n − k) of all orthonormal k -frames in R n. The isotropy representation of this homogeneous space contains equivalent summands, so a complete description of G -invariant metrics is not easy. In this paper we view the manifold V 2 p R n as total space over a classical generalized flag manifold with two isotropy summands and prove for 2 ≤ p ≤ 2 5 n − 1 it admits at least four invariant Einstein metrics determined by Ad (U (p) × SO (n − 2 p)) -invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics. The Einstein equation reduces to a parametric system of polynomial equations, which we study by combining Gröbner bases and geometrical arguments. [ABSTRACT FROM AUTHOR]

In the first part, using the recent measure classification results of Eskin-Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space G/Γ. Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of Simmons-Weiss, we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so, in particular, well approximable. [ABSTRACT FROM AUTHOR]

We review known and state some new results on the homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of the homotopy nilpotency of homogeneous spaces G/K for a Lie group G and its closed subgroup K ă G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)), Ω(Fn;n1,...,nk (K)), and Ω(Vn,m(K)) of Grassmann Gn,m(K), flag Fn;n1,...,nk (K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is studied. [ABSTRACT FROM AUTHOR]

In this paper, we associate a growth graph to a homogeneous space of a compact group. Under certain assumptions, we show that the spectral dimension of a homogeneous space is greater than or equal to summability of the length operator associated with the growth graph. Using this, we compute spectral dimension of spheres. Communicated by Miriam Cohen [ABSTRACT FROM AUTHOR]

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space Rd∖{0}. We construct simple adapted tight frames for L2(Rd) that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous α‐modulation spaces is introduced. [ABSTRACT FROM AUTHOR]

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was investigated and positively answered in previous papers. It is conjectured that this result can be improved, namely that any homogeneous Finsler manifold admits at least two homogenous geodesics. Examples of homogeneous Randers manifolds admitting just two homogeneous geodesics are presented. [ABSTRACT FROM AUTHOR]

We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient M ≃ G / K {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds. [ABSTRACT FROM AUTHOR]

There exists a holomorphic quadratic differential defined on any H-surface immersed in the homogeneous space E(κ t) given by U. Abresch and H. Rosenberg, called the Abresch-Rosenberg differential. However, there was no Codazzi pair on such an H-surface associated with the Abresch-Rosenberg differential when t ≠ 0. The goal of this paper is to find a geometric Codazzi pair defined on any H-surface in E(κ, τ), when t ≠ 0, whose (2, 0)-part is the Abresch-Rosenberg differential. We denote such a pair as (I, IIAR), were I is the usual first fundamental form of the surface and IIAR is the Abresch-Rosenberg second fundamental form. In particular, this allows us to compute a Simons' type equation for H-surfaces in E(κ, τ). We apply such Simons' type equation, first, to study the behavior of complete H-surfaces S of finite Abresch-Rosenberg total curvature immersed in E(κ, τ). Second, we estimate the first eigenvalue of any Schrodinger operator L = Δ+V, V continuous, defined on such surfaces. Finally, together with the Omori-Yau maximum principle, we classify complete H-surfaces in E(κ, τ), τ ≠ 0, satisfying a lower bound on H depending on κ, τ, and an upper bound on the norm of the traceless IIAR, a gap theorem. [ABSTRACT FROM AUTHOR]

We introduce the class of θ n -Urysohn spaces and the n - θ -closure operator. θ n -Urysohn spaces generalize the notion of a Urysohn space and we consider their relationship with S(n)-spaces, studied in [9], [14] and [18]. We estabilish bounds on the cardinality of these spaces and cardinality bounds if the space is additionally homogeneous. [ABSTRACT FROM AUTHOR]

Using the characterization of the spin representation in terms of exterior forms, we give a complete classification of invariant spinors on the nine homogeneous realizations of the sphere S n. In each of the cases we determine the dimension of the space of such spinors, give their explicit description, and study the underlying related geometric structures depending on the metric. We recover some known results in the Sasaki and 3-Sasaki cases and find several new examples: in particular we give the first known examples of generalized Killing spinors with four distinct eigenvalues. [ABSTRACT FROM AUTHOR]

The general question considered below is: how the properties of a homogeneous compact space X are influenced by the fact that X contains a "nice" closed subspace F which is also "nicely" located in X ?. A typical result says that if the tightness of F is countable, and X ∖ F is metalindelöf, then the tightness of X is also countable. A question posed by Oleg Pavlov is answered. Some applications of the results obtained to perfect maps are given. The concept of G δ -homogeneous space is introduced, and it is shown, in particular, that if X is a G δ -homogeneous compact space with countable tightness, then the weight of X does not exceed 2 ω (Corollary 3.8). [ABSTRACT FROM AUTHOR]

Abstract In this paper we show that every naturally reductive space can be obtained by an explicit construction. This allows us to write a general formula for any naturally reductive space from which we prove a reducibility and isomorphism criteria. [ABSTRACT FROM AUTHOR]