Narrow Results

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a 'small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on 'stability' and 'local rigidity' of discontinuous groups. As a test case, we give an explicit description of the deformation space of ℤ^k acting properly discontinuously on ℝ^k+1 by affine nilpotent transformations. Our method uses an idea of 'continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle E_H → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad(E_H) is homogeneous. Fix a faithful H-module V. We also show that E_H is homogeneous if the vector bundle E_H ×^H V associated to it for the H-module V is homogeneous.

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad(K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E₆/(SU(4) × SU(2) × U(1) × U(1)) and E₇/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2ℓ + 1)/(U(1) × U(p) × SO(2(ℓ - p - 1) + 1)) and SO(2ℓ)/(U(1) × U(p) × SO(2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

We define and study generalized homogeneous Besov spaces connected with the Riemann–Liouville operator. We establish some results of density of subspaces, completeness and continuous embedding. Also, a discrete equivalent norm is examined.

We define a Larotonda space as a quotient space ${𝒫}={{𝒰}}_{{𝒜}}/{{𝒰}}_{{ℬ}}$ of the unitary groups of $C^{\ast }$-algebras $1\in {ℬ}\subset {𝒜}$ with a faithful unital conditional expectation

Given a point $\rho \in {𝒫}$ and a tangent vector $X\in {(T{𝒫})}_{\rho }$, we consider the problem of whether the geodesic $\delta$ of the linear connection satisfying these initial data is (locally) minimal for the metric. We find a sufficient condition. Several examples are given, of locally minimal geodesics.

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of a compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. Then we introduce the abstract notions of convolution and involution over the Banach measure space $M(G/H)$.

We study geometric variational problems for a class of nonlinear σ-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces, we obtain a weaker result on existence of minimizers in each 2-homotopy class.

Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G → G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.

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.

The central limit theorem (CLT) for stationary ergodic Markov chains is investigated. We give a short survey of related results on the CLT for general (not necessarily Harris recurrent) chains and formulate a new sufficient condition for its validity. Furthermore, Markov operators are considered which admit invariant orthogonal splittings of the space of square-integrable functions. We show how conditions for the CLT can be improved if this additional structure is taken into account. Finally we give examples of this situation, namely endomorphisms of compact Abelian groups and random walks on compact homogeneous spaces.

There is a unipotent subgroup of Sl(n, C) whose action on the flag manifold of Sl(n, C) completes the flows of the complex Kostant–Toda lattice (a Hamiltonian system in Lax form) through initial conditions where all the eigenvalues coincide. The action preserves the Bruhat cells, which are in one-to-one correspondence with the elements of the permutation group Σ_n. A generic orbit in a given cell is homeomorphic to C^m, where m is determined by the "gap sequence" of the permutation, which lists the number inversions of each degree.

Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space . The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto.

Four-dimensional pseudo-Riemannian homogeneous spaces whose isotropy is non-trivial with commuting curvature operators have been studied. The only example of homogeneous Einstein four-manifold which is curvature-Ricci commuting but not semi-symmetric has been presented. Non-trivial examples of semi-symmetric homogeneous four-manifolds which are not locally symmetric, also Jacobi–Jacobi commuting manifolds which are not flat have been presented.

In this paper, we consider invariant $(\alpha ,\beta )$-metrics which are induced by invariant Riemannian metrics $ã$ and invariant vector fields $\tilde{X}$ on homogeneous spaces. We study the flag curvatures of invariant $(\alpha ,\beta )$-metrics. We first give an explicit formula for the flag curvature of invariant $(\alpha ,\beta )$-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.

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.

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.

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a paper by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.

We obtain new invariant Einstein metrics on the compact Lie group SU(2n) (n ≥ 3) which are not naturally reductive. We view the compact Lie group SU(2n) as total space over the generalized Wallach space SU(2n)/U(n) and consider Ad(U(n))-invariant metrics on the group SU(2n) associated to the generalized Wallach space to obtain these invariant Einstein metrics.

We obtain new invariant Einstein metrics on the compact Lie groups Sp(n) which are not naturally reductive. This is achieved by using the quaternionic flag manifolds Sp(k₁ + … + k_p)/Sp(k₁) × … × Sp(k_p) and by imposing certain symmetry assumptions in the set of all left-invariant metrics on Sp(n).

We consider invariant Einstein-like metrics of type ℬ, that is, Ricci tensor r is a Codazzi tensor, on generalized flag manifolds. We show that, for generalized flag manifolds G/K with second Betti number b₂(G/K) = 2 and with ν isotropy summands (ν ≤ 5) or G₂-type, then invariant Einstein-like metrics of type ℬ are Einstein.

For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature invariants of these manifolds vanish.