Compact Homogeneous Spaces Research Articles

Sub-Riemannian geometries on compact homogeneous spaces

We consider totally non-holonomic sub-Riemannian structures on compact homogeneous spaces and conjecture that, in most cases, the existence of such a structure implies that there is an invariant Riemannian structure. We prove a strengthened version of this conjecture for some important classes of compact homogeneous spaces.

On invariant sub-Riemannian structures on compact homogeneous spaces with discrete stationary subgroup

In the paper, invariant distributions on compact homogeneous spaces with discrete stationary subgroup are studied. Special attention is paid to the presence of a sub-Riemannian structure on these distributions and to the condition of complete nonholonomicity. A description of the above homogeneous spaces admitting an invariant completely nonholonomic sub-Riemannian structure is obtained.

Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers

We prove that the multiplication maps $\mathbb{s}^n \times \mathbb{s}^n \rightarrow \mathbb{s}^n$ ($n = 1, 3, 7$) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range, the unique Lipschitz constant minimizers in their homotopy classes. Other geometrically natural maps, such as projections of Hopf fibrations, have already been shown to be, up to isometries, the unique Lipschitz constant minimizers in their homotopy classes, and it is suspected that this may hold true for all Riemannian submersions of compact homogeneous spaces.

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.

The classification of homogeneous Einstein metrics on flag manifolds with [formula omitted

Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Let $M$ be a compact Riemannian homogeneous space (e.g., a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation $\mathbf{D}_{t}\partial_{t}u=\sum_{k=1}^{d}\mathbf{D}_{x_{k}}\partial_{x_{k}}u+f_{u}(Du)+g_{u}(Du)\dot{W}$ in any dimension $d\ge1$, where $f$ and $g$ are continuous multilinear maps, and $W$ is a spatially homogeneous Wiener process on $\mathbb{R}^{d}$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.

Invariant distributions on compact homogeneous spaces

In this paper, we study distributions on compact homogeneous spaces, including invariant distributions and also distributions admitting a sub-Riemannian structure. We first consider distributions of dimension 1 and 2 on compact homogeneous spaces. After this, we study the cases of compact homogeneous spaces of dimension 2, 3, and 4 in detail. Invariant distributions on simply connected compact homogeneous spaces are also treated.Bibliography: 18 titles.

Random walks and approximate integration on compact homogeneous spaces

We discuss methods of producing random walks on a compact homogeneous space $X$ and examine how they lead to approximate evaluation of integrals of elements of various function spaces, including $L^{p}$ spaces, $L^{p}$-Sobolev spaces, and Hölder spaces.

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.

Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

Let \(M=G/H\) be a compact connected isotropy irreducible Riemannian homogeneous manifold, where \(G\) is a compact Lie group (may be, disconnected) acting on \(M\) by isometries. This class includes all compact irreducible Riemannian symmetric spaces and, for example, the tori \(\mathbb{R }^n/\mathbb{Z }^n\) with the natural action on itself extended by the finite group generated by all permutations of the coordinates and inversions in circle factors. We say that \(u\) is a polynomial on \(M\) if it belongs to some \(G\)-invariant finite dimensional subspace \(\mathcal{E }\) of \(L^2(M)\). We compute or estimate from above the averages over the unit sphere \(\mathcal{S }\) in \(\mathcal{E }\) for some metric quantities such as Hausdorff measures of level set and norms in \(L^p(M)\), \(1\le p\le \infty \), where \(M\) is equipped with the invariant probability measure. For example, the averages over \(\mathcal{S }\) of \(\Vert u\Vert _{L^p(M)}\), \(p\ge 2\), are less than \(\sqrt{\frac{p+1}{e}}\) independently of \(M\) and \(\mathcal{E }\).

Weakly complex homogeneous spaces

Abstract. We complete our recent classification (2011) of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann (1955)), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer (1986)), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.

Nonnegatively curved homogeneous metrics in low dimensions

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, we examine all $G$-invariant fibration metrics on $G/H$ for $G$ a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of nonnegative sectional curvature.

Global Quantization of Pseudo-Differential Operators on Compact Lie Groups, SU(2), 3-sphere, and Homogeneous Spaces

Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hormander's classes of operators $\Psi^m(G)$, $\Psi^m(S^3)$ and $\Psi^m(SU(2))$. Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as $L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global quantization are established on general compact Lie groups.

Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds

Given a Stein manifold XC which is homogeneous under a complex reductive Lie group GC, i.e., a complexification GC/KC of a compact homogeneous space G/K. Consider a relatively compact domain D which is invariant w.r.t. the compact real form G of the complex reductive Lie group in the Stein manifold XC. We find a relation between the automorphism group of the invariant domain D and isometric group of the compact homogeneous space G/K. When the compact homogeneous space G/K is isotropy irreducible, or even more general, we obtain a rigidity property of the automorphism groups.

Classification of compact homogeneous spaces with invariant G2-structures

Classification of compact homogeneous spaces with invariant G2-structures

Toric Genera of Homogeneous Spaces and their Fibrations

The aim of this paper is to study further the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus $T^k$. As the torus action in this case has only isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In that context for homogeneous $SU$-spaces we prove the well known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any $T^k$-equivariant Hirzebruch genus given by an odd power series vanishes. Related to the problem of multiplicativity we provide construction of stable complex $T^k$-fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces.

Double quiver gauge theory and nearly Kähler flux compactifications

We consider G-equivariant dimensional reduction of Yang-Mills theory with torsion on manifolds of the form MxG/H where M is a smooth manifold, and G/H is a compact six-dimensional homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures which includes a nearly Kahler structure. We establish an equivalence between G-equivariant pseudo-holomorphic vector bundles on MxG/H and new quiver bundles on M associated to the double of a quiver Q, determined by the SU(3)-structure, with relations ensuring the absence of oriented cycles in Q. When M=R^2, we describe an equivalence between G-invariant solutions of Spin(7)-instanton equations on MxG/H and solutions of new quiver vortex equations on M. It is shown that generic invariant Spin(7)-instanton configurations correspond to quivers Q that contain non-trivial oriented cycles.

Classification of compact homogeneous spaces with invariant G2-structures

In this note we classify all homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$ acts effectively on $G/H$. They include a subclass of all homogeneous spaces $G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and low rigidity and show that they admit families of invariant coclosed $G_2$-structures (resp. $\tilde G_2$-structures).

Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces

We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schrodinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

Invariant measures and controllability of finite systems on compact manifolds

A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of (Ph. Jouan, ESAIM: COCV 16 (2010) 956-973). and of the existence of an invariant measure on certain compact homogeneous spaces.