Flows with periodic factors on homogeneous spaces

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.

We compute the Borel equivariant cohomology ring of the left K-action on a homogeneous space G=H , where G is a connected Lie group, H and K are closed connected subgroups, and 2 as well as the torsion primes of the Lie groups are units of the coefficient ring.As a special case, this gives the singular cohomology rings of biquotients H n G=K.This depends on a version of the Eilenberg-Moore theorem developed in the appendix, where a novel multiplicative structure on the two-sided bar construction B.A 00 ; A; A 0 / is defined, valid when A 00A !A is a pair of maps of homotopy Gerstenhaber algebras.16U80, 55N91, 57T15, 57T30; 16E45, 55T20, 57T35 Homogeneous spaces, which can be realized as coset spaces G=H for G a transitively acting Lie group and H the stabilizer of a point, are arguably the most highly symmetric, most canonical, and most thoroughly investigated objects of study in differential geometry after Lie groups.A generalization of perennial interest, which offers many interesting examples in positive-curvature geometry, is the class of biquotients, the orbit spaces KnG=H of G under free left-right actions by products K H of two closed subgroups.The cohomology rings of a Lie group G over Q and those finite fields F p for which H .GI Z/ lacks p-torsion have been known to be exterior algebras since the fundamental 1941 work of Hopf [17], and that of a homogeneous space G=H with connected stabilizer H has been known over R since work of Henri Cartan from 1950 [9]: H .G=H / Tor H .BG/ .R; H .BH//: In his 1952 dissertation [6, Section 30], Borel used the Serre spectral sequence of the Borel fibration G ! G=H !BH to show the same ring isomorphism also holds when H is of maximal rank, with F p coefficients if H .GI Z/ and H .H I Z/ lack p-torsion and with Z coefficients if they are torsion free.Beyond this, however, ring isomorphisms proved hard to find.Starting with Baum's 1962 dissertation, a program began to obtain a more general result, additively at least, using the then-new Eilenberg-Moore spectral sequence of the fibration G=H !BH !BG.Cartan's result implies this spectral sequence's collapse over R, and Baum proved collapse under hypotheses covering the best-studied homogeneous spaces.Subsequent work of authors including Baum [2], Larry Smith [30], Gugenheim and May [15],

Let Γ \Gamma and Γ ′ \Gamma ’ be lattices, and ϕ \phi and ϕ ′ \phi ’ one-parameter subgroups of the connected Lie groups G G and G ′ G’ . If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces G / Γ G/\Gamma and G ′ / Γ ′ G’ /\Gamma ’ are topologically equivalent, then they are topologically equivalent by an affine map. (a) G G and G ′ G’ are one-connected and nilpotent. (b) G G and G ′ G’ are one-connected and solvable, and for all X X in L ( G ) L(G) and X ′ X’ in L ( G ′ ) L(G’ ) , ad ⁡ ( x ) \operatorname {ad} (x) and ad ⁡ ( X ′ ) \operatorname {ad} (X’ ) have only real eigenvalues, (c) G G and G ′ G’ are centerless and semisimple with no compact direct factor and no direct factor H H isomorphic to PSL ⁡ ( 2 , R ) \operatorname {PSL} (2,\,R) such that Γ H \Gamma H is closed in G G . Moreover, in condition (c), the induced flow of ϕ \phi on G / Γ G/\Gamma is assumed to be ergodic. Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say ϕ \phi is isolated if and only if for any ϕ ′ \phi ’ which recurrently approaches ϕ \phi for positive and negative time, ϕ \phi equals ϕ ′ \phi ’ up to sense-preserving reparameterization. Theorem B(a) states that if G G is one-connected and nilpotent, or one-connected and solvable with exp: L ( G ) → G L(G) \to G a diffeomrophism, then every ϕ \phi of G G is isolated. Let G G be connected and semisimple and ϕ ( t ) = exp ⁡ ( t X ) \phi (t) = \exp (tX) . Then Theorem B(b) states that ϕ \phi is isolated, if [ X , Y ] = 0 [X,\,Y] = 0 and ad ⁡ ( Y ) \operatorname {ad} (Y) being semisimple imply that ad ⁡ ( Y ) \operatorname {ad} (Y) has some eigenvalue not pure imaginary and not zero. If G G has finite center, ϕ \phi is isolated if there is no compact connected subgroup in the centralizer of ϕ \phi .

Smooth embeddings of SL(2) and PGL(2)

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.

Reductive homogeneous Lorentzian manifolds

We introduce the notion of the \({\mathfrak{g}}{\mathfrak{l}(V)}\)-prolongation of Lie algebras of differential operators on homogeneous spaces. The \({\mathfrak{g}}{\mathfrak{l}(V)}\)-prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where V is a homogeneous space. We apply the obtained results to the spaces S1 (the Virasoro algebra) and \({\mathbb{R}}^1 \).

LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and\(\overline {GA} = L\). LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U+ ={gεG:a −n gan} →e as n → ∞. Let Ω be a non-empty open subset ofU + andn i → ∞ be any sequence. It is showed that\(\overline { \cup _{i = 1}^\infty a^n \Omega \Lambda } = L\). A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.

Integrable theories in any dimension and homogenous spaces

Le K be a compact connected Lie group, L be a connected closed subgroup of K. It is well known that L is a subgroup of maximal rank of K if and only if the Euler characteristic of the manifold M = K/L is positive. Such homogeneous spaces M have been classified in [7, 10]. However, their topological classification was unknown. This classification is obtained in the present article. We show tha two compact homogeneous spaces M = K/L and M′ = K′/L′ of positive Euler characteristic are diffeomorphic if and only if the graded rings H *(M,Z) and H *(M′,Z) are isomorphic. We also obtain the rational homotopy classification of such homogeneous spaces which is not equivalent to the differential one. These results were announced in [15].

Given a compact, connected Lie group G with Lie algebra \(\mathfrak{g}\). We discuss time-optimal control of bilinear systems of the form $$\dot U(t) = \left( {H_d + \sum\limits_{j = 1}^m {v_j (t)H_j } } \right)U(t),$$ ((I)) where H d , H j ∈ \(\mathfrak{g}\), U ∈ G, and the v j act as control variables. The case G = SU(2n) has found interesting applications to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n-particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the system from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote by the Lie algebra spanned by H 1, ..., H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G/K we discuss in detail time-optimal control strategies for system (I) in the case where G/K carries the structure of a Riemannian symmetric space.

We show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

In this article, we develop the algorithm for finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford–Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford–Klein forms. By a numerical experiment, we show that there is a large class of homogeneous spaces satisfying Tholozan’s condition.

The study of geodesic flows on homogenous spaces is an area of research that has yielded some fascinating developments. This book, first published in 2000, focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture. Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders. The treatment is as self-contained and elementary as possible. It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory

We discuss manifolds with positive sectimal curvature on which a group acts isometrically with one dimensional quotient. A number of the known examples have this property, but some potential families for new examples in dimension 7 arise as well. We discuss the geometry of these known examples and the connection that the candidates have with self-dual Einstein metrics. There are very few known examples of manifolds with positive sectional curvature. Apart from the compact rank one symmetric spaces, they exist only in dimensions 24 and below and are all obtained as quotients of a compact Lie group equipped with a biinvariant metric under an isometric group action. They consist of certain homogeneous spaces in dimensions 6,7,12,13, and 24 due to Berger [Be], Wallach [Wa], and Aloff–Wallach [AW], and of biquotients in dimensions 6,7, and 13 due to Eschenburg [E1],[E2],[E3] and Bazaikin [Ba]. When trying to find new examples, it is natural to search among manifolds with large isometry group, a program initiated by K. Grove in the 1990s, see [Wi] for a recent survey. Homogeneous spaces with positive curvature were classified in [Be],[Wa],[BB] in the 1970s. The next natural case to study is therefore manifolds on which a group acts isometrically with one dimensional quotient, so called co-homogeneity one manifolds. L. Verdiani [V1, V2] showed that in even dimensions, positively curved cohomogeneity one manifolds are equivariantly diffeomorphic to an isometric action on a rank one symmetric space. In odd dimensions, K. Grove and the author observed in 1998 that there are infinite families among the known nonsymmetric positively curved manifolds that admit isometric cohomogeneity one actions, and suggested a family of potential 7 dimensional candidates P k . In [GWZ], a classification in odd dimensions was carried out, and another family Q k and an isolated manifold R emerged in dimension 7. It is not yet known whether these manifolds admit a cohomogeneity one metric with positive curvature, although they all admit one with nonnegative curvature as a consequence of the main result in [GZ]. In [GWZ], the authors also discovered an intriguing connection that the manifolds P k and Q k have with a family of self-dual Einstein orbifold metrics constructed by Hitchin [Hi1] on S4. They naturally give rise to 3-Sasakian metrics on P k and Q k , which by definition have lots of positive curvature already. The purpose of this survey is threefold. In Section 2, we study the positively curved cohomogeneity one metrics on known examples with positive curvature, including the explicit functions that define the metric. In Section 3, we describe the classification theorem in [GWZ]. It is remarkable that among 7-manifolds where G = S3 ×S3 acts by cohomogeneity one, one has the known positively curved Eschenburg spaces E p , the Berger space B 7, the Aloff–Wallach space W 7, and the sphere S7, and that the candidates P k , Q k , and R all carry such an action as well. We thus carry out the proof in this most intriguing case where G = S3 ×S3 acts by cohomogeneity one on a compact 7-dimensional simply connected manifold. In Section 4, we describe the relationship to Hitchin’s self-dual Einstein metrics. We also discuss some curvature properties of these Einstein metrics and the metrics they define on P k and Q k . The behavior of these metrics, as well as the known metrics with positive curvature, are illustrated in a series of pictures.