Homogeneous Riemannian Research Articles

Homogeneous Riemannian Structures in Thurston Geometries and Contact Riemannian Geometries

We give explicit parametrizations for all the homogeneous Riemannian structures on model spaces of Thurston geometry. As an application, we give all the homogeneous contact metric structures on $3$-dimensional Sasakian space forms.

Almost Global Asymptotic Trajectory Tracking for Fully-Actuated Mechanical Systems on Homogeneous Riemannian Manifolds

Almost Global Asymptotic Trajectory Tracking for Fully-Actuated Mechanical Systems on Homogeneous Riemannian Manifolds

Standard homogeneous (α1,α2)${\big(\alpha _1,\alpha _2\big)}$‐metrics and geodesic orbit property

AbstractIn this paper, we introduce the notion of standard homogeneous ‐metric, as a generalization for the normal homogeneity in Finsler geometry with relatively good computability. We explore the geodesic orbit (g.o. in short) property of this metric. Especially, when it is associated with a triple of compact Lie groups, we find new examples of g.o. Finsler spaces from H. Tamaru's classification list. Meanwhile, we prove that all standard g.o. ‐metrics on the Wallach spaces, , and , must be the normal homogeneous Riemannian metrics.

Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces \((M=G/H,g)\) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups \({{\,\mathrm{SO}\,}}(n)\), \(\mathrm{U}(n)\), and H is a diagonally embedded product \(H_1\times \cdots \times H_s\), where \(H_j\) is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M=G∕H,g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (Sp(n)∕Sp(n1)×⋯×Sp(ns),g), with 0<n1+⋯+ns≤n. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces (G∕H,g) with H semisimple.

Motion of Charged Particle in a Class of Homogeneous Spaces

We study the motion of charged particle under a natural choice of electromagnetic field in a general class of compact homogeneous spaces. As a special case we describe the motion in homogeneous Riemannian spaces $(G/H,g)$, where $g$ is any deformation of a normal metric along the fibers of a homogeneous fibration $K/H\rightarrow G/H \rightarrow G/K$.

On the geometrical properties of Heisenberg groups

In [20] the existence of major differences about totally geodesic two-dimensional foliations between Riemannian and Lorentzian geometry of the Heisenberg group $H_{3}$ is proved. Our aim in this paper is to obtain a comparison on some other geometrical properties of these spaces. Interesting behaviours are found. Also the non-existence of left-invariant Ricci and Yamabe solitons and the existence of algebraic Ricci soliton in both Riemannian and Lorentzian cases are proved. Moreover, all of the descriptions of their homogeneous Riemannian and Lorentzian structures and their types are obtained. Besides, all the left-invariant generalized Ricci solitons and unit time-like vector fields which are spatially harmonic are completely determined.

On the Geometry of Higher Dimensional Heisenberg Groups

In this paper, we first completely determine all left-invariant generalized Ricci solitons on the Heisenberg group \(H_{2n+1}\) equipped with any left-invariant Riemannian and Lorentzian metric that this Lie group admits. Then, we explicitly calculate the energy of an arbitrary left-invariant vector field V on these spaces and in the Lorentzian cases we determine the exact form of all left-invariant unit time-like vector fields which are spatially harmonic. We also obtain all of the descriptions of their homogeneous Riemannian and Lorentzian structures and explicitly distinguish their types. Finally, we investigate parallel hypersurfaces of these spaces and show that these spaces never admit any totally geodesic hypersurface. The existence of algebraic Ricci solitons and the non-existence of left-invariant Ricci solitons and Yamabe solitons on these spaces in both Riemannian and Lorentzian cases is proved. Also, different behaviors regarding the existence of harmonic maps, critical points for the energy functional restricted to vector fields and some equations in Riemannian and Lorentzian cases are found.

Применение пакетов символьных вычислений к исследованию алгебраических солитонов Риччи на однородных (псевдо)римановых многообразиях

Применение пакетов символьных вычислений к исследованию алгебраических солитонов Риччи на однородных (псевдо)римановых многообразиях

Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

In this paper, we discuss generalizations of the classical Bianchi–Backlund transformation of two-dimensional pseudo-spherical surfaces in three-dimensional spaces of constant curvature to the case of submanifolds of arbitrary codimension in spaces of constant curvature and in homogeneous Riemannian products.

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.

Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martínez and A. Verjovsky on the minimality of this flow assuming that the “natural” affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G.A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, which places us within the homogeneous framework investigated by M. Ratner. Since our purpose is to deal with non-homogeneous situations, we do not use Ratner’s famous Orbit-Closure Theorem, but we give an elementary proof. We show that this special situation arises for homogeneous Riemannian and Lie foliations, reintroducing the foliation point of view. Examples and counter-examples take an important place in our work, in particular, the very instructive case of the solvable manifold TA3. Our aim in writing this text is to offer to the reader an accessible introduction to a subject that was intensively studied in the algebraic setting, although there still are unsolved geometric problems.

Reduction of Homogeneous Riemannian Structures

AbstractThe goal of this paper is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group H of isometries. In a first result, H is a normal subgroup of the group of symmetries associated with the reducing tensor . The situation when H is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fibrings of almost contact manifolds over almost Hermitian manifolds. If the structure is, moreover, Sasakian, the obtained reduced tensor is homogeneous Kähler.

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres \({S^n}\). We prove that for any connected (almost effective) transitive on \(S^n\) compact Lie group \(G\), the family of \(G\)-invariant Riemannian metrics on \(S^n\) contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and \(n\ge 5\). Any such family (that exists only for \(n=2k+1\)) contains a metric \(g_\mathrm{can}\) of constant sectional curvature \(1\) on \(S^n\). We also prove that \((S^{2k+1}, g_\mathrm{can})\) is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to \(G\) (except the groups \(G={ SU}(k+1)\) with odd \(k+1\)). The space of unit Killing vector fields on \((S^{2k+1}, g_\mathrm{can})\) from Lie algebra \(\mathfrak g \) of Lie group \(G\) is described as some symmetric space (except the case \(G=U(k+1)\) when one obtains the union of all complex Grassmannians in \(\mathbb{C }^{k+1}\)).

Generalized normal homogeneous spheres

We find new generalized normal homogeneous but not normal homogeneous Riemannian metrics on spheres of dimensions 4n+3, n ≥ 1, and all homogeneous space forms covered by them; all these spaces have zero Euler characteristic. Deriving consequences, alongside some other new results we obtain new proofs for analogous known results for all complex projective spaces of odd complex dimension starting from three.

Minimal surfaces in $\widetilde{\mathit{PSL}_{2}(\mathbb{R})}$

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{\mathit{PSL}_{2}(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions defined over bounded domains of the hyperbolic plane.

Homogeneous structures on real and complex hyperbolic spaces

The connected groups acting by isometries on either the real or the complex hyperbolic spaces are determined. A Lie-theoretic description of the homogeneous Riemannian, respectively Kahler, structures of linear type on these spaces is then found. On both spaces, examples that are not of linear type are given.

Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

We state a Wiener criterion for the regularity of points with respect to a relaxed Dirichlet problem for a p -homogeneous Riemannian Dirichlet form.