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 non-negative curvature is maintained for small deformations. Building on the work of Schwachhöfer and Tapp (J. Geom. Anal. 19(4):929–943, 2009), 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 non-negative sectional curvature.     

Homogeneous Metrics with Nonnegative Curvature

Given compact Lie groups H⊂G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.     

Homogeneous almost normal Riemannian manifolds

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Löwner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.     

Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups

Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S³ = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S³ of all geodesics on (G, d) with respect to p.     

CLASSIFICATION OF FINITE-MULTIPLICITY SYMMETRIC PAIRS

We give a complete classification of the reductive symmetric pairs (G, H) for which the homogeneous space (G × H)/ diag H is real spherical in the sense that a minimal parabolic subgroup has an open orbit. Combining with a criterion established in T. Kobayashi, T. Oshima, Adv. Math. 2013, we give a necessary and sufficient condition for a reductive symmetric pair (G, H) such that the multiplicities for the branching law of the restriction of any admissible smooth representation of G to H have finiteness/boundedness property.     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Stability of Discontinuous Groups Acting on Homogeneous Spaces

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ,G,H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ,H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ,G) and study the case of Heisenberg groups.     

The geometry of compact homogeneous spaces with two isotropy summands

We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on T _p M decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup H <  K <  G, we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by Manturov (Dokl. Akad. Nauk SSSR 141, (1961), 792–795 1034–1037, Tr. Semin. Vector Tensor Anal. 13, (1966), 68–145) and J Wolf (Acta Math. 120, (1968), 59–148 152, (1984) 141–142).      

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

Left Lie reduction for curves in homogeneous spaces

Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : G → G/H is a Riemannian submersion with respect to a bi-invariant Riemannian metric on G. Given a C^∞ curve x : [a, b] → G/H, let \(\tilde {x}:[a,b]\rightarrow G\) be the horizontal lifting of x with \(\tilde {x}(a)=e\), where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction\(V(t):=\tilde x(t)^{-1}\dot {\tilde x}(t)\) of \(\dot {\tilde x}(t)\) for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector \(\dot {x}(t)\) from x(t) to x(a) along x. Then left Lie reductions are used to investigate Riemannian cubics, Riemannian cubics in tension and elastica in homogeneous spaces G/H. Simplifications of reduced equations are found when (G, H) is a Riemannian symmetric pair. These equations are compared with equations known for curves in Lie groups, focusing on the special case of Riemannian cubics in the 3-dimensional unit sphere S³.     

Classification of homogeneous almost cosymplectic three-manifolds

The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R×N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.     

Detecting Hilbert manifolds among isometrically homogeneous metric spaces

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H, where G is a metrizable topological group and H is a closed balanced subgroup of G.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a tessellation if there is a discrete subgroup of G, such that acts properly discontinuously on G/H, and the double-coset space \G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples.It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, Kulkarni and Kobayashi constructed examples that are not obvious when G=SO(2, 2n)° or SU(2, 2n). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G=SO(2, 2n)°. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G=SU(2, 2n). This includes the construction of another family of examples.The main results are obtained from methods of Benoist and Kobayashi: we fix a Cartan decomposition G=K A ⁺ K, and study the intersection (K H K)A ⁺. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.     

Obstruction to Positive Curvature on Homogeneous Bundles

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F) were discovered recently [J. Differential Geom. 65:273–287, 2003; Invent. Math. 148:117–141, 2002]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h) × F with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F = {point} proven by Geroch [Proc. Amer. Math. Soc. 66(2):321–326, 1977].Supported in part by NSF grant DMS–0303326.     

On weakly S-embedded and weakly τ-embedded subgroups

Let G be a finite group. A subgroup H of G is said to be weakly S-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H _seG , where H _seG is the subgroup generated by all those subgroups of H which are S-quasinormally embedded in G. We say that a subgroup H of G is weakly τ-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H _seG , where H _seG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. In this paper, we study the properties of weakly S-embedded and weakly τ-embedded subgroups, and use them to determine the structure of finite groups.     

Einstein homogeneous bisymmetric fibrations

We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K/L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.     

Orbits and Invariants Associated with a Pair of Spherical Varieties: Some Examples

Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H=K is a symmetric subgroup of G, then the double coset space K\G/K (and the corresponding invariant theoretic quotient) are no longer finite, but several nice properties hold, including an analogue of the Chevalley restriction theorem. These properties were generalized by Helminck and Schwarz (Duke Math. J. 106(2) (2001), pp. 237–279) to the case where H and K are fixed point groups of commuting involutions. We recall Helminck and Schwarz's main results. We also give examples to show the difficulty in extending these results if we allow H=K to be a reductive spherical (nonsymmetric) subgroup or if we have H symmetric and K spherical reductive.     

Proper smooth G-manifolds have complete G-invariant Riemannian metrics

Assume G is a Lie group, K is a compact subgroup of G and M is a proper smooth G-manifold. Using properties of the regular representations L²(G) and L²(K), we first prove results about extending certain representations and embedding homogeneous spaces smoothly into Hilbert G-spaces. We then prove that M can be embedded as a closed smooth G-invariant submanifold of some Hilbert G-space. It follows that M admits a complete G-invariant smooth Riemannian metric.