Geodesic orbit manifolds and Killing fields of constant length

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We prove the following result: If $\mathfrak{a}$ is an abelian ideal of $\mathfrak{g}$, then every Killing field $X\in \mathfrak{a}$ has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space.Comment: 7 pages, some typos are correcte

Negative eigenvalues of the Ricci operator of solvable metric Lie algebras

In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra has this property.Comment: 16 pages, minor correction

Classification of generalized Wallach spaces

In this paper, we present the classification of generalized Wallach spaces and discuss some related problems.Comment: 18 pages, 1 figur

Killing vector fields of constant length on compact homogeneous Riemannian manifolds

In this paper we present some structural results on the Lie algebras of transitive isometry groups of a general compact homogenous Riemannian manifold with nontrivial Killing vector fields of constant length.Comment: 26 pages, small revision, accepted for publication in Annals of Global Analysis and Geometr

On invariant Riemannian metrics on Ledger-Obata spaces

We study invariant metrics on Ledger-Obata spaces $F^m/\mathrm{diag}(F)$. We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case $m=3$, any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger-Obata space is reducible if and only if it is isometric to the product of Ledger-Obata spaces (and give an effective method of recognising reducible metrics), and that the full connected isometry group of an irreducible Ledger-Obata space $F^m/\mathrm{diag}(F)$ is $F^m$. We deduce that a Ledger-Obata space is a geodesic orbit manifold if and only if it is the product of naturally reductive Ledger-Obata spaces.Comment: 18 page

The signature of the Ricci curvature of left invariant Riemannian metrics on 4-dimensional Lie groups

In this paper, we present the classification of all possible signatures of the Ricci curvature of left-invariant Riemannian metrics on 4-dimensional Lie groups and discuss some related questions.Comment: 12 pages, extended and corrected versio

Solvable Lie groups of negative Ricci curvature

We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, and discuss some open questions.Comment: 18 page

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

Generalized Popoviciu's problem

This is an English translation of the following paper, published several years ago: Nikonorov Yu.G., Nikonorova Yu.V. Generalized Popoviciu's problem (Russian), Tr. Rubtsovsk. Ind. Inst., 7, 229-232 (2000), Zbl. 0958.51021. All inserted footnotes provide additional information related to the mentioned problem.Comment: 5 pages, 1 figur

Geodesic orbit Riemannian spaces with two isotropy summands. I

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. The main result is the classification of compact simply connected geodesic orbit Riemannian spaces $G/H$ with two irreducible submodules in the isotropy representation.Comment: 15 pages, comments are welcom