Locally homogeneous Riemannian manifolds and Cartan triples

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.

Download Full-text

Curvature-orbits and locally homogeneous riemannian manifolds

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf01765845 ◽

1993 ◽

Vol 165 (1) ◽

pp. 121-131 ◽

Cited By ~ 7

Author(s):

Federico G. Lastaria ◽

Franco Tricerri

Keyword(s):

Riemannian Manifolds ◽

Homogeneous Riemannian Manifolds ◽

Locally Homogeneous

Download Full-text

On strictly locally homogeneous Riemannian manifolds

Differential Geometry and its Applications ◽

10.1016/s0926-2245(96)00043-5 ◽

1997 ◽

Vol 7 (2) ◽

pp. 131-137

Author(s):

Oldřich Kowalski

Keyword(s):

Riemannian Manifolds ◽

Homogeneous Riemannian Manifolds ◽

Locally Homogeneous

Download Full-text

Investigation of conformally killing vector fields on 5-dimensional 2-symmetric lorentzian manifolds

Yugra State University Bulletin ◽

10.17816/byusu20210117-22 ◽

2021 ◽

Vol 60 (1) ◽

pp. 17-22

Author(s):

Tatiana A. Andreeva ◽

Dmitry N. Oskorbin ◽

Evgeny D. Rodionov

Keyword(s):

Riemannian Manifolds ◽

Symmetric Spaces ◽

Vector Fields ◽

Killing Vector ◽

Lorentzian Manifolds ◽

Killing Vector Fields ◽

Homogeneous Riemannian Manifolds ◽

Locally Homogeneous ◽

Riemannian Case ◽

Riemannian Symmetric Spaces

Conformally Killing fields play an important role in the theory of Ricci solitons and also generate an important class of locally conformally homogeneous (pseudo) Riemannian manifolds. In the Riemannian case, V. V. Slavsky and E.D. Rodionov proved that such spaces are either conformally flat or conformally equivalent to locally homogeneous Riemannian manifolds. In the pseudo-Riemannian case, the question of their structure remains open. Pseudo-Riemannian symmetric spaces of order k, where k 2, play an important role in research in pseudo-Riemannian geometry. Currently, they have been investigated in cases k=2,3 by D.V. Alekseevsky, A.S. Galaev and others. For arbitrary k, non-trivial examples of such spaces are known: generalized Kachen - Wallach manifolds. In the case of small dimensions, these spaces and Killing vector fields on them were studied by D.N. Oskorbin, E.D. Rodionov, and I.V. Ernst with the helpof systems of computer mathematics. In this paper, using the Sagemath SCM, we investigate conformally Killing vector fields on five-dimensional indecomposable 2- symmetric Lorentzian manifolds, and construct an algorithm for their computation.

Download Full-text

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

Izvestiya of Altai State University ◽

10.14258/izvasu(2021)1-15 ◽

2021 ◽

pp. 93-96

Author(s):

D.V. Vylegzhanin ◽

P.N. Klepikov ◽

O.P. Khromova

Keyword(s):

Riemannian Manifolds ◽

Three Dimensional ◽

Riemannian Metric ◽

Isotropy Subgroup ◽

Lorentzian Manifolds ◽

Ricci Operator ◽

Homogeneous Riemannian Manifolds ◽

Left Invariant Riemannian Metric ◽

Locally Homogeneous ◽

Unique Eigenvalue

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.

Download Full-text