Oldřich Kowalski

On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.

Geodesics in weakly symmetric spaces

We prove that any maximal geodesic in a weakly symmetric space is an orbit of a one-parameter group of isometries of that space.

On Tangent Sphere Bundles with Small or Large Constant Radius

For a Riemannian manifold M, we determine somecurvature properties of a tangent sphere bundleTrM endowed with the induced Sasaki metric in the case when the constantradius r > 0 of the tangent spheres is either sufficientlysmall or sufficiently large.

D'Atri spaces

Riemannian symmetric spaces play an important role in many fields of mathematics and physics. They have been studied by several generations of mathematicians and from different viewpoints. Their classification is well-known. Usually they are characterized as Riemannian manifolds whose geodesic symmetries (that is, geodesic reflections with respect to all points) are globally defined isometries. Hence, these geodesic symmetries are also volume-preserving. This observation led D’Atri and Nickerson to the study of Riemannian and pseudo-Riemannian manifolds all of whose (local) geodesic symmetries are volume-preserving (up to sign) or equivalently, which are divergence-preserving. They are introduced in [DN1], [DN2], [D] where the first non-symmetric examples, namely all naturally reductive homogeneous spaces, are found and where various other characterizations are given. Following [VW1] such spaces are called D’Atri spaces.

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].

Counter-example to the “Second Singer's theorem”

We give an explicit example showing that a theorem by I. M. Singer announced in [3] (about the existence of a Riemannian homogeneous space with the prescribed curvature tensor and some of its covariant derivatives) cannot hold without an additional topological condition of closeness.All references in this short note concern Chapter 3 of the paper by L. Nicolodi and F. Tricerri [2] published in the same volume. We shall use freely the concepts and formulas from there.Consider the infinitesimal model (V,T,K) given as follows: LetV be a 5-dimensional

On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

We find the necessary and sufficient conditions for three constants ϱ1, ϱ2, ϱ3 ∈ ℝ3 to be the principal Ricci curvatures of some 3-dimensional locally homogeneous Riemannian space.

Non-homogeneous relatives of symmetric spaces

Abstract For every fixed Riemannian symmetric space ( M , g ) we determine explicitly all locally non-homogeneous Riemannian spaces which have, at all points, the same curvature tensor as ( M , g ). For this purpose, we describe explicitly all parabolically foliated semi-symmetric spaces in the sense of Z.I. Szabo.

On Locally Nonhomogeneous Pseudo–Riemannian Manifolds with Locally Homogeneous Levi–Civita Connections

In this paper we make the first steps to a classification of (pseudo-) Riemannian manifolds which are not locally homogeneous but their Levi–Civita connections are homogeneous. The full classification is given for dimension n = 2; in higher dimensions we prove some substantial partial results. In more generality, we are also interested in the difference between the dimension of the algebra of affine Killing vector fields and that of the algebra of metric Killing vector fields (without any homogeneity properties).

The classification of ?-symmetric Sasakian manifolds

A ϕ-symmetric spaceM is a complete connected regular Sasakian manifold, that fibers over an Hermitian symmetric spaceN, so that the geodesic involutions ofN lift to define global (involutive) automorphisms of the Sasakian structure onM. In the present paper the complete classification of ϕ-symmetric spaces is obtained. The groups of automorphisms of the Sasakian structures and the groups of isometries of the underlying Riemannian metrics are determined. As a corollary, the Sasakian space forms are also determined.