Riemannian Homogeneous Spaces Research Articles

Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

AbstractWe study hypersurfaces of the four‐dimensional Thurston geometry , which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of . We also give a closed expression for the Riemann curvature tensor of , using two integrable complex structures.

Compact geodesic orbit spaces with a simple isotropy group

Let M=G/H be a compact, simply connected, Riemannian homogeneous space, where G is (almost) effective and H is a simple Lie group. In this paper, we first classify all G-naturally reductive metrics on M, and then all G-geodesic orbit metrics on M.

The Ambrose–Singer Theorem for General Homogeneous Manifolds with Applications to Symplectic Geometry

The main Theorem of this article provides a characterization of reductive homogeneous spaces equipped with some geometric structure (not necessarily pseudo-Riemannian) in terms of the existence of certain connection. This result generalizes the well-known Theorem of Ambrose and Singer for Riemannian homogeneous spaces (Ambrose and Singer in Duke Math J 25(4):647–669, 1958). We relax the conditions in this theorem and prove a characterization of reductive locally homogeneous manifolds. Finally, we apply these results to classify, with explicit expressions, reductive locally homogeneous almost symplectic, symplectic and Fedosov manifolds.

Geodesic orbit spaces of compact Lie groups of rank two

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H, g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.

On Weakly Reflective Submanifolds in Compact Isotropy Irreducible Riemannian Homogeneous Spaces

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

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 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.

Algebraic properties of bounded Killing vector fields

In this paper, we consider a connected Riemannian manifold $M$ where a connected Lie group $G$ acts effectively and isometrically. Assume $X\in\mathfrak{g}=\mathrm{Lie}(G)$ defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition $X=X_r+X_s$ according to a Levi decomposition $\mathfrak{g}=\mathfrak{r}(\mathfrak{g})+\mathfrak{s}$, where $\mathfrak{r}(\mathfrak{g})$ is the radical, and $\mathfrak{s}=\mathfrak{s}_c\oplus\mathfrak{s}_{nc}$ is a Levi subalgebra. The decomposition $X=X_r+X_s$ coincides with the abstract Jordan decomposition of $X$, and is unique in the sense that it does not depend on the choice of $\mathfrak{s}$. By these properties, we prove that the eigenvalues of $\mathrm{ad}(X):\mathfrak{g}\rightarrow\mathfrak{g}$ are all imaginary. Furthermore, when $M=G/H$ is a Riemannian homogeneous space, we can completely determine all bounded Killing vector fields induced by vectors in $\mathfrak{g}$. We prove that the space of all these bounded Killing vector fields, or equivalently the space of all bounded vectors in $\mathfrak{g}$ for $G/H$, is a compact Lie subalgebra, such that its semi-simple part is the ideal $\mathfrak{c}_{\mathfrak{s}_c}(\mathfrak{r}(\mathfrak{g}))$ of $\mathfrak{g}$, and its Abelian part is the sum of $\mathfrak{c}_{\mathfrak{c}(\mathfrak{r}(\mathfrak{g}))} (\mathfrak{s}_{nc})$ and all two-dimensional irreducible $\mathrm{ad}(\mathfrak{r}(\mathfrak{g}))$-representations in $\mathfrak{c}_{\mathfrak{c}(\mathfrak{n})}(\mathfrak{s}_{nc})$ corresponding to nonzero imaginary weights, i.e. $\mathbb{R}$-linear functionals $\lambda:\mathfrak{r}(\mathfrak{g})\rightarrow \mathfrak{r}(\mathfrak{g})/\mathfrak{n}(\mathfrak{g}) \rightarrow\mathbb{R}\sqrt{-1}$, where $\mathfrak{n}(\mathfrak{g})$ is the nilradical.

VolterraNet: A Higher Order Convolutional Network With Group Equivariance for Homogeneous Manifolds.

Convolutional neural networks have been highly successful in image-based learning tasks due to their translation equivariance property. Recent work has generalized the traditional convolutional layer of a convolutional neural network to non-euclidean spaces and shown group equivariance of the generalized convolution operation. In this paper, we present a novel higher order Volterra convolutional neural network (VolterraNet) for data defined as samples of functions on Riemannian homogeneous spaces. Analagous to the result for traditional convolutions, we prove that the Volterra functional convolutions are equivariant to the action of the isometry group admitted by the Riemannian homogeneous spaces, and under some restrictions, any non-linear equivariant function can be expressed as our homogeneous space Volterra convolution, generalizing the non-linear shift equivariant characterization of Volterra expansions in euclidean space. We also prove that second order functional convolution operations can be represented as cascaded convolutions which leads to an efficient implementation. Beyond this, we also propose a dilated VolterraNet model. These advances lead to large parameter reductions relative to baseline non-euclidean CNNs. To demonstrate the efficacy of the VolterraNet performance, we present several real data experiments involving classification tasks on spherical-MNIST, atomic energy, Shrec17 data sets, and group testing on diffusion MRI data. Performance comparisons to the state-of-the-art are also presented.

Convergence of locally homogeneous spaces

We study three different topologies on the moduli space $$\mathcal {H}^\mathrm{loc}_m$$ of equivariant local isometry classes of m-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed $$\mathcal {C}^{k,\alpha }$$ -topology, for some $$k>1$$ , which do not admit any convergent subsequence in the pointed $$\mathcal {C}^{k+1}$$ -topology.

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 $K/H\rightarrow G/H \rightarrow G/K$.

Convergence of Lie group integrators

We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.

Equiaffine Connections on Three-Dimensional Pseudo-Riemannian Spaces

The question of description equiaffine connections on a smooth manifold is studied. In general, the purpose of the research is quite complicated. Therefore, it is natural to consider this problem in a narrower class of pseudo-Riemannian manifolds, for example, in the class of homogeneous pseudo-Riemannian manifolds. In this paper for all three-dimensional Riemannian and pseudo-Riemannian homogeneous spaces, it is determined under what conditions the connection is equiaffine (locally equiaffine). In addition, equiaffine (locally equiaffine) connections, torsion tensors and Ricci tensors are written out in explicit form.

Some Notes Concerning Riemannian Submersions and Riemannian Homogenous Spaces

Riemannian submersions between Lie groups and Riemannian homogeneous spaces are investigated. With the help of connections, some characterizations dealing these spaces are obtained.

Statistics on the Stiefel manifold: Theory and applications

A Stiefel manifold of the compact type is often encountered in many fields of engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces. In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Fréchet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based nonrecursive counter part as well as the stochastic gradient descent based method prevalent in literature.

Изотропный тензор Вейля на четырехмерных локально однородных псевдоримановых многообразиях

The locally homogeneous (pseudo)Riemannian manifolds are studied by many mathematicians. The more general cases of these manifolds are spaces where conformal transformations act transitively. They are also called the locally conformally homogeneous (pseudo)Riemannian manifolds. It is worth noting that such manifolds were investigated both for the case of Riemannian metric, and for the case of pseudo-Riemannian metric. The work of E.D. Ro dionov, V.V. Slavskii, and L.N. Chibrikova claims that if there is a Weyl tensor with the non-zero squared length for a manifold of dimension n ≥ 4, then a locally homogeneous space can be obtained from a locally conformally homogeneous (pseudo)Riemannian space by means of a conformal deformation. Hence, there should be a naturally arisen problem to investigate such (pseudo)Riemannian locally conformally homogeneous and locally homogeneous manifolds for which the squared length of the Weyl tensor equals to zero, but the tensor itself is not equal to zero (such a Weyl tensor is also called isotropic). In this paper, we describe a step-by-step algorithm to solve the problem of classification of four-dimensional locally homogeneous pseudo-Riemannian manifolds with an isotropic Weyl tensor and a non-trivial isotropy subgroup.

On homogeneous manifolds whose isotropy actions are polar

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek–Ricci spaces have non-polar isotropy actions.

Analogs of the Globevnik problem on Riemannian two-point homogeneous spaces

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.

Алгебраические солитоны Риччи на метрических группах Ли с недиагонализируемым оператором Риччи

In recent years, various generalizations of Einstein manifolds are actively studied, for example, manifolds with the trivial Schouten-Weyl tensor, and Ricci solitons, which were first considered by R. Hamilton. Ricci solitons on homogeneous (pseudo)Rieman-nian spaces and, in particular, on the Lie groups have been studied by many mathematicians. For example, there are no nontrivial homogeneous invariant Ricci solitons on three and four-dimensional Lie groups with a left-invariant Riemannian metric. A similar result was proved for the unimodular Lie groups with a left-invariant Riemannian metric in any dimension. However, this question is still an open problem for nonunimodular Lie groups of dimension more than 4. Another important example of Ricci solitons is the case of algebraic Ricci solitons on Lie groups, first considered by J. Lauret. Later, it was proved that every algebraic Ricci soliton on a Lie group with left-invariant (pseudo)Riemannian metric is a homogeneous Ricci soliton. This paper shows the existence of non-trivial algebraic and homogeneous invariant Ricci solitons on conformally flat Lie groups in the case of nondiagonalizable Ricci operator. Also, a non-diagonalizable Ricci operator on Lie groups with harmonic Weyl tensor is demonstrated.

Towards the classification of odd-dimensional homogeneous reversible Finsler spaces with positive flag curvature

In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd-dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that most important features of L. Berard-Bergery’s classification results for odd-dimensional positively curved Riemannian homogeneous spaces can be generalized to reversible Finsler spaces.