Homogeneous pseudo-Riemannian Manifolds Research Articles

Eigenvalues of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with a Four-Dimensional Isotropy Subgroup

The problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator was studied in the papers of many mathematicians. This problem was solved by O. Kowalski and S. Nikcevic for the case of three-dimensional locally homogeneous Riemannian manifolds. The work of G. Calvaruso and O. Kowalski contains the answer to the question above for the case of three –dimensional locally homogeneous Lorentzian manifolds.
 For the four-dimensional case, similar studies were carried out only in the case of Lie groups with a left-invariant Riemannian metric. The works of A.G. Kremlyov and Yu.G. Nikonorov presented the possible signatures of the eigenvalues of the Ricci operator. However, the question of recovering a four-dimensional Lie group with a left-invariant Riemannian metric from a given Ricci operator remains open.
 This paper is devoted to the study of the eigenvalues of the Ricci operator on four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a four-dimensional isotropy subgroup. An algorithm for calculating the eigenvalues of the Ricci operator is presented. A theorem on the restoration of such manifolds from a given Ricci operator is proved. It is established that such possibility can happen only in the case when the prescribed operator is diagonalizable and has a unique eigenvalue of multiplicity four.

Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Given a homogeneous pseudo-Riemannian space $$(G/H,\langle \ , \ \rangle),$$ a geodesic $$\gamma :I\rightarrow G/H$$ is said to be two-step homogeneous if it admits a parametrization $$t=\phi (s)$$ (s affine parameter) and vectors X, Y in the Lie algebra $${\mathfrak{g}}$$ , such that $$\gamma (t)=\exp (tX)\exp (tY)\cdot o$$ , for all $$t\in \phi (I)$$ . As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics $$\langle \ ,\ \rangle$$ on the unimodular Lie group $$SL(2,{{\mathbb{R}}})$$ such that $$\big (SL(2,{{\mathbb{R}}}),\langle \ ,\ \rangle \big )$$ is a two-step g.o. space.

Mathematical Modeling in the Study of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with Isotropic Weyl Tensor

It is known that a locally homogeneous manifold can be obtained from a locally conformally homogeneous (pseudo)Riemannian manifolds by a conformal deformation if the Weyl tensor (or the Schouten-Weyl tensor in the three-dimensional case) has a nonzero squared length. Thus, the problem arises of studying (pseudo)Riemannian locally homogeneous and locally conformally homogeneous manifolds, the Weyl tensor of which has zero squared length, and itself is not equal to zero (in this case, the Weyl tensor is called isotropic).
 One of the important aspects in the study of such manifolds is the study of the curvature operators on them, namely, the problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator.
 The problem of the prescribed values of the Ricci operator on 3-dimensional locally homogeneous Riemannian manifolds has been solved by O. Kowalski and S. Nikcevic. Analogous results for the one-dimensional and sectional curvature operators were obtained by D.N. Oskorbin, E.D. Rodionov, and O.P Khromova.
 This paper is devoted to the description of an example of studying the problem of the prescribed Ricci operator for four-dimensional locally homogeneous (pseudo) Riemannian manifolds with a nontrivial isotropy subgroup and isotropic Weyl tensor.

On four-dimensional locally homogeneous pseudo-Riemannian manifolds with isotropic Weyl tensor

On four-dimensional locally homogeneous pseudo-Riemannian manifolds with isotropic Weyl tensor

Parallel and totally geodesic hypersurfaces of non‐reductive homogeneous four‐manifolds

AbstractWe classify totally geodesic and parallel hypersurfaces of four‐dimensional non‐reductive homogeneous pseudo‐Riemannian manifolds.

On locally homogeneous pseudo-Riemannian compact Einstein manifolds

We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford–Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein Riemannian compact manifolds using T. Kobayashi’s conjecture as a guiding principle.

The almost complex (para-complex) structures on 6-pseudo-Riemannian spheres and related Schrödinger flows

In this paper, by using the $$G_{2(2)}$$ -structure on $$\hbox {Im}(\mathbf{Ca'})\cong {\mathbb {R}}^{3,4}$$ of the purely imaginary Cayley’s split-octaves $$\mathbf{Ca'}$$ , the $$G_{2(2)}$$ -bi-normal motion of curves $$\gamma _t(s)$$ in the pseudo-Euclidean space $${\mathbb {R}}^{3,4}$$ is studied. The motion is closely related to Schrodinger flows into homogeneous pseudo-Riemannian manifolds $${\mathbb {S}}^{2,4}(1)=G_{2(2)}/SU(1,2)$$ of signature (2, 4) and $${\mathbb {S}}^{3,3}(\sqrt{-1})=G_{2(2)}/SL(3,{\mathbb {R}})$$ of signature (3, 3), in which $${\mathbb {S}}^{2,4}(1)$$ (resp. $${\mathbb {S}}^{3,3}(\sqrt{-1}))$$ admits the almost complex (resp. para-complex) structure. Furthermore, the motion of spacelike (resp. timelike) curves is also shown to be equivalent to the nonlinear Schrodinger-like system (resp. the nonlinear coupled heat equations) in three unknown functions, which generalizes the correspondence between the bi-normal motion of timelike (resp. spacelike) curves in $${\mathbb {R}}^{2,1}$$ and the defocusing nonlinear Schrodinger equation (resp. the nonlinear heat equation). To show this correspondence, $$G_{2(2)}$$ -frame field on the curve is used.

Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces

The aim of this work is to study homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces. First, we deduce a formula for Ricci tensor of a homogeneous pseudo-Riemannian manifold with compact isotropy subgroup. Based on this formula, we establish a one-to-one correspondence between homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces and homogeneous Riemannian Einstein metrics on compact homogeneous spaces. As an application, we prove that every noncompact connected simple Lie group except SL(2) admits at least two nonproportional left invariant pseudo-Riemannian Einstein metrics. Furthermore, we study left invariant pseudo-Riemannian Einstein metrics on solvable Lie groups. By showing that if a nilpotent Lie group admits a left invariant Riemannian Ricci soliton, then it admits a left invariant pseudo-Riemannian Ricci soliton as well, we construct a left invariant pseudo-Riemannian Einstein metric on any given Riemannian standard Einstein solvmanifold.

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.

On Classification of Four-dimensional Locally Homogeneous Pseudo-Riemannian Manifolds with an Isotropic Weyl Tensor

We investigate four-dimensional locally homogeneous pseudo-Riemannian manifolds with an isotropic Weyl tensor. An algorithm for obtaining a complete classification of such manifolds is described.

О классификации четырехмерных локально однородных псевдоримановых многообразий с изотропным тензором Вейля

О классификации четырехмерных локально однородных псевдоримановых многообразий с изотропным тензором Вейля

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.

SPIN STRUCTURES ON COMPACT HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

Исследование четырехмерных локально однородных (псевдо)римановых многообразий с изотропным тензором Схоутена — Вейля

Локально однородные (псевдо)римановы многообразия изучались в работах многих математиков. Их обобщением являются локально конформно однородные (псевдо)римановы пространства, на которых транзитивно действуют конформные преобразования. Такие многообразия также ранее исследовались как в римановом случае, так и в псевдоримановом.В работе Е.Д. Родионова, В.В. Славского и Л.Н. Чибриковой было доказано, что из локально конформно однородного (псевдо)риманова пространства можно с помощью конформной деформации получить локально однородное пространство, если тензор Вейля (или тензор Схо-утена — Вейля в трехмерном случае) имеет ненулевой квадрат длины. Таким образом возникает задача об изучении (псевдо)римановых локально однородных и локально конформно однородных многообразий, тензор Схоутена — Вейля которых имеет нулевой квадрат длины, а сам не равен нулю.В данной работе приводится алгоритм, с помощью которого можно решить задачу о классификации четырехмерных локально однородных (псевдо)римановых многообразий с нетривиальной подгруппой изотропии и изотропным тензором Схоутена — Вейля.DOI 10.14258/izvasu(2018)4-14

Four-dimensional pseudo-Riemannian g.o. spaces and manifolds

Abstract A g.o. manifold is a homogeneous pseudo-Riemannian manifold whose geodesics are all homogeneous, that is, they are orbits of a one-parameter group of isometries. A g.o. space is a realization of a homogeneous pseudo-Riemannian manifold ( M , g ) as a coset space M = G ∕ H , such that all the geodesics are homogeneous. We prove that apart from the already classified non-reductive examples (Calvaruso et al., 2015), any four-dimensional pseudo-Riemannian g.o. manifold is naturally reductive. To obtain this result, we shall also provide a complete description up to isometries of four-dimensional pseudo-Riemannian g.o. spaces, and show explicit realizations of the four-dimensional pseudo-Riemannian naturally reductive spaces classified in Batat et al. (2015).

Application of Computer Mathematics Systems to the Study of Homogeneous (pseudo)Riemannian Manifolds with the Trivial Schouten — Weyl Tensor

Application of Computer Mathematics Systems to the Study of Homogeneous (pseudo)Riemannian Manifolds with the Trivial Schouten — Weyl Tensor

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

Применение пакетов символьных вычислений к исследованию оператора одномерной кривизны на нередуктивных однородных псевдоримановых многообразиях

The study of curvature operator properties, in particular, the one-dimensional curvature operator, is interesting for the understanding of the geometrical and topological structure of a homogeneous (pseudo)Riemannian manifold. In general case, this problem is quite difficult. Therefore, it is necessary to impose restrictions either on the class of manifolds or their dimension. An application of analytical calculation systems is possible if the dimension is finite. In this paper, mathematical and computer models for determining the components of the one-dimensional curvature operator and its spectrum (the set of eigenvalues) of non-reductive homogeneous (pseudo)Riemannian manifolds of a finite dimension are developed. The investigation of one-dimensional curvature operator spectrum on non-reductive homogeneous Lorentzian manifolds of dimension 4 is performed by implementing this algorithm in Maple software. Also, a symmetric operator with a matrix corresponding to a matrix of the one-dimensional curvature tensor is defined, and the problem of this operator possible signature existence on four-dimensional non-reductive homogeneous Lorentzian manifolds is studied.

On locally homogeneous compact pseudo-Riemannian manifolds

On locally homogeneous compact pseudo-Riemannian manifolds

Conformally Flat Pseudo-riemannian Homogeneous Ricci Solitons 4-spaces

We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.