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.     

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.     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

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.     

The Chebyshev norm on the lie algebra of the motion group of a compact homogeneous Finsler manifold

In this paper, we prove that the natural metric on the connected component of the unit in the (Lie) motion group of a compact Finsler manifold supplied with its inner metric generates a bi-invariant inner Finsler metric. The latter is defined by the invariant Chebyshev norm on the Lie algebra of generators of 1-parameter motion subgroups on the manifold. This norm is equal to the maximal value of the generator’s length. A δ-homogeneous manifold is characterized by the condition that the canonical projection of the component onto the manifold is a submetry with respect to their inner metrics. The Chebyshev norms for the Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on the 3-dimensional complex projective space are found. This gives interesting examples of invariant norms on Lie algebras and a new method for the separating of delta-homogeneous but not normal metrics. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

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.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces

Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space ?(Γ,G,H) is a Hausdorff space, where Γ acts on the threadlike homogeneous space G/H as a discontinuous subgroup. Whenever k = rank(Γ) > 3, this space is also shown to be endowed with a smooth manifold structure. But if k = 3, then ?(Γ,G,H) admits a smooth manifold structure as its open dense subset. These phenomena are strongly linked to the features of adjoint orbits of the basis group G on the parameter space ?(Γ,G,H) (which is semi-algebraic in this case) and specifically to their dimensions, as it will be seen throughout the paper. This also allows to provide a proof of the Local Rigidity Conjecture in this setup.     

On coset-spaces of compact Lie groups by subgroups of corank 2

Let G be a simple compact connected simply connected Lie group, H its connected Lie subgroup of corank 2 which coincides with the commutator group of the centralizer of a torus, and let Sam(G/H) = 0. We prove that if a compact connected simply connected Lie group G' acts transitively and locally effectively on the manifold G/H, then G' is isomorphic to G. if the root system of G consists of roots of the same length, then the action of G' on G/H is similar to the action of G.     

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.     

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.     

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.     

A Canonical Connection for Lie Group Actions

The aim of this paper is to generalize the construction of an Ambrose-Singer connection for Riemannian homogeneous manifolds to the case of cohomogeneity one Riemannian manifolds. Necessary and sufficient conditions are given on a Riemannian manifold (M,g) in order that there exists a Lie group of isometries acting on M with principal orbits of codimension one.     

Geodesic symmetries of homogeneous Kähler Manifolds

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kähler metric onG/T does not have volume-preserving geodesic symmetries. From the Kähler/de Rham decomposition of a compact homogeneous Kähler manifold [8], our result extends to the invariant Kähler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kähler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G^C of G on. X, this lifts to a holomorphic action of G^C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G^C of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.     

The Taylor map on complex path groups

Let W(G) denote the path group of an arbitrary complex connected Lie group. The existence of a heat kernel measure νt on W(G) has been shown in [M. Cecil, B.K. Driver, Heat kernel measure on loop and path groups, preprint, http://www.math.uconn.edu/~cecil/papers/p2.pdf; Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted for publication]. The present work establishes an isometric map, the Taylor map, from the space of L²(νt)-holomorphic functions on W(G) to a subspace of the dual of the universal enveloping algebra of Lie(H(G)), where H(G) is the Lie subgroup of finite energy paths. This map is shown to be surjective in the case where G is a simply connected graded Lie group.     

The Cyclomatic Number of a Graph and its Independence Polynomial at −1

If by s _k is denoted the number of independent sets of cardinality k in a graph G, then ${I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}$ is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97–106, 1983), where α = α(G) is the size of a maximum independent set. The inequality |I (G; ?1)| ≤ 2 ^ν(G), where ν(G) is the cyclomatic number of G, is due to (Engström in Eur. J. Comb. 30:429–438, 2009) and (Levit and Mandrescu in Discret. Math. 311:1204–1206, 2011). For ν(G) ≤ 1 it means that ${I(G;-1)\in\{-2,-1,0,1,2\}.}$ In this paper we prove that if G is a unicyclic well-covered graph different from C ₃, then ${I(G;-1)\in\{-1,0,1\},}$ while if G is a connected well-covered graph of girth ≥ 6, non-isomorphic to C ₇ or K ₂ (e.g., every well-covered tree ≠ K ₂), then I (G; ?1) = 0. Further, we demonstrate that the bounds {?2 ^ν(G), 2 ^ν(G)} are sharp for I (G; ?1), and investigate other values of I (G; ?1) belonging to the interval [?2 ^ν(G), 2 ^ν(G)].     

A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.     

Magnetic trajectories in three-dimensional Lie groups

We study magnetic trajectories in Lie groups equipped with bi-invariant Riemannian metric. We define the Lorentz force of a magnetic field in a Lie group G, and then, we give the Lorentz force equation for the associated magnetic trajectories that are curves in G. When the manifold is a Lie group G equipped with bi-invariant Riemannian metric, we derive differential equation system that characterizes magnetic flow associated with the Killing magnetic curves with regard to the Lie reduction of the curve γ in G.