On a class of compact homogeneous spaces II

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

This paper presents a systematic study for trigonometric polynomials over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of a compact group $G$. Using the abstract notion of dual space $\\widehat{G/H}$, we introduce the space of trigonometric polynomials $\\mathrm{Trig}(G/H)$ over the compact homogeneous space $G/H$. As an application for harmonic analysis of trigonometric polynomials, we prove that the abstract dual space of anyhomogeneous space of compact groups separates points of the homogeneous space in some sense.

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c ⋅ g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.

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.

Let G be a compact connected Lie group, K a closed subgroup (not necessarily connected) and M = G/K the homogeneous space of left cosets. Assume that M is orientable and p * : H n ( G ) → H n ( M ) is nonzero, where n = dim M . In this paper, we employ an equivariant version of Nielsen root theory to show that the converse of the Lefschetz fixed-point theorem holds true for all selfmaps on M . Moreover, if the Lefschetz number of a selfmap f : M → M is nonzero, then the Nielsen number of f coincides with the Reidemeister number of f , which can be computed algebraically.

The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones. This construction is based on the notion of a certain duality between compact and non-compact homogeneous spaces. In the second part we apply this method to obtain pseudo-Riemannian homogeneous manifolds with real Killing spinors. We will prove that under a certain additional condition a dual pseudo-Riemannian space (G′/H′, g′) of a compact Riemannian homogeneous space (G/H, g) with homogeneousSpin-structure admits a homogeneousSpin+-structure and theG_invariant Killing spinors on (G/H, g) correspond toG′-invariant Killing spinors on (G′/H′, g′). We can ensure that in most cases the hypothesis onG-invariance is satisfied.

Introduction .220 Chapter I. A fixed point theorem for finite diffeomorphism groups generated by reflections 221 Chapter II. Geometric weight system and a fixed point theorem for compact connected Lie groups.... 231 1. Actions of compact simple Lie groups of rank 1.232 2. Basic properties of the geometric weight system.237 3. A fixed point theorem for compact connected Lie groups.240 Chapter III. Classification of the types of principal isotropy subgroups for actions of compact connected Lie groups on acyclic manifolds .242 1. Classification of principal orbit types for actions of simple compact connected Lie groups on acyclic manifolds .243 2. Classification of principal orbit types for actions of general compact connected Lie groups .247 References ...... 255

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.

Let $\alpha_0$ be an affine action of a discrete group $\Gamma$ on a compact homogeneous space $X$ and $\alpha_1$ a smooth action of $\Gamma$ on $X$ which is $C^1$-close to $\alpha_0$. We show that under some conditions, every topological conjugacy between $\alpha_0$ and $\alpha_1$ is smooth. In particular, our results apply to Zariski-dense subgroups of $SL_d(\mathbb{Z})$ acting on the torus $\mathbb{T}^d$ and Zariski-dense subgroups of a simple noncompact Lie group $G$ acting on a compact homogeneous space $X$ of $G$ with an invariant measure.

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.

Compact Lie groups and homogeneous spaces

Preface.- Programme of the Workshop on Boundaries.- Programme of the Alp-Workshop 2009.- Publications of D.I. Cartwright.- Publications of M.A. Picardello.- Publications of V.A. Kaimonvich.- M.J. Dunwoody, An Inaccessible Graph.- J. Parkinson and B. Schapira, A Local Limit Theorem for Random Walks on the Chambers of A2 Buildings.- A. Erschler, On Continuity of Range, Entropy and Drift for Random Walks on Groups.- Y. Guivarc'h and C.R.E. Raja, Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces.- M. Bjoerklund, Ergodic Theorems for Homogeneous Dilations.- A. Gnedin, Boundaries from Inhomogeneous Bernoulli Trials.- P.E.T. Jorgensen and E.P.J. Pearse, Resistance Boundaries of Infinite Networks.- M. Arnaudon and A. Thalmaier, Brownian Motion and Negative Curvature.- R.K. Wojciechowski, Stochastically Incomplete Manifolds and Graphs.- S. Haeseler and M. Keller, Generalized Solutions and Spectrum for Dirichlet Forms on Graphs.- R. Froese, D. Hasler and W. Spitzer, A Geometric Approach to Absolutely Continuous Spectrum for Discrete Schroedinger Operators.- A. Bendikov, B. Bobikau and C. Pittet, Some Spectral and Geometric Aspects of Countable Groups.- P. Muller and P. Stollmann, Percolation Hamiltonians.- T.S. Turova, Survey of Scalings for the Largest Connected Component in Inhomogeneous Random Graphs.- D. D'Angeli, A. Donno and T. Nagnibeda, Partition Functions of the Ising Model on Some Self-similar Schreier Graphs.- I. Krasovsky, Aspects of Toeplitz Determinants.

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.

The operators \( \mathfrak{U}_\delta\) which we studied in Chapter 3 have analogues in the compact case. In this chapter we study their properties for compact two-point homogeneous spaces of dimension > 1. These are the Riemannian manifolds M with the property that for any two pairs of points \((p_1,\,p_2)\,{\rm and}\,(q_1,q_2)\,{\rm satisfying}\, d((p_1,\,p_2)\,=\,(q_1,q_2)\,{\rm where}\, d{\rm \,is\, the \,distance \,on}\, M\), there exists an isometry mapping \((p_1\,to\,p_2)\,{\rm and}\,(p_1\,to\,p_2).\) By virtue of Wang’s classification (see Helgason [H5, Chapter 1, § 4]) these are also the compact symmetric spaces of rank one. Unlike the non-compact case, the treatment in this chapter is based on the realizations of the spaces under consideration. Accordingly, the use of Lie theory is minimal.KeywordsSymmetric SpaceHarmonic PolynomialRiemannian Symmetric SpaceComplex QuadraticWiener TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper is a survey of results (partly obtained by the authors) on homogeneous spaces of Lie groups \(G\) with a compact stabilizer subgroup \(H\), on which every \(G\)-invariant distribution is integrable. It is proved that the condition of integrability is necessary and sufficient for every invariant inner metric to be (holonomic) Finsler on such a space. As a corollary of the obtained results, we assert that the class of homogeneous spaces with invariant non-holonomic Riemannian metrics (in other terms, sub-Riemannian or Carnot–Caratheodory metrics), which were actively studied last 3 decades after Gromov’s work, is rather broad. On the other hand, the class of homogeneous spaces with integrable invariant distributions includes Cartan’s symmetric spaces as well as isotropy irreducible, in particular, strictly isotropy irreducible, homogeneous spaces, which have been classified in simply connected case in the papers by Wang and Ziller (respectively, by Manturov, Wolf and Kramer). Special attention is paid to the case, when the Lie groups \(G\) and \(H\) are connected. Then the integrability condition of the invariant distributions is equivalent to a purely algebraic condition, that for the Lie algebra \(h\) of the subgroup \(H\), any \(ad(h)\)-invariant vector subspace in the Lie algebra \(g\) of the Lie group \(G\) is a Lie subalgebra; such Lie subalgebra \(h\subset g\) is called a strong subalgebra. The first author proved that a simply connected and compact space \(G/H\) with this condition is isomorphic to a direct product of strictly isotropy irreducible homogeneous spaces. In line with this, the second author recently found several non-compact simply connected homogeneous spaces with this condition, which are not isomorphic to such direct products. These results are naturally related to the structure questions of a class of general homogeneous locally compact spaces with an inner metric. This class is exactly the closure in the Gromov–Hausdorff sense of the class of homogeneous manifolds with an inner metric. Any such manifold is isometric to some homogeneous manifold \(G/H\) with \(G\)-invariant (may be, non-holonomic) Finsler metric. The authors give fairly detailed survey of the existing methods of the search of geodesics, i.e., locally shortest arcs, on such manifolds (particularly, with invariant non-holonomic Riemannian metrics), non-holonomic metric geometry and its relations with the geometric group theory, \(CR\)-manifolds, thermodynamics, etc. Some unsolved problems are suggested.