A remark on locally homogeneous Riemannian spaces

A locally homogeneous Riemannian space is called non-regular if it is not locally isometric to any globally homogeneous Riemannian space. We show that no non-regular space has non positive Ricci tensor and that a theorem by Alkseevski-Kimelfeld may be extended to the class of locally homogeneous spaces: i.e. any locally homogeneous Riemannian space with zero Ricci tensor is locally euclidean.     

On similarity homogeneous locally compact spaces with intrinsic metric

In this article, we generalize partially the theorem of V. N. Berestovskii on characterization of similarity homogeneous (nonhomogeneous) Riemannian manifolds, i.e., Riemannian manifolds admitting transitive group of metric similarities other than motions to the case of locally compact similarity homogeneous (nonhomogeneous) spaces with intrinsic metric satisfying the additional assumption that the canonically conformally equivalent homogeneous space is δ-homogeneous or a space of curvature bounded below in the sense of A. D. Aleksandrov. Under the same assumptions, we prove the conjecture of V. N. Berestovskii on topological structure of such spaces.     

Locally conformally homogeneous pseudo-Riemannian spaces

Locally homogeneous Riemannian spaces were studied in [1–4]. Locally conformally homogeneous Riemannian spaces were considered in [10]. Moreover, the theorem claiming that every such space is either conformally flat or conformally equivalent to a locally homogeneous Riemannian space was proved.In this article, we study locally conformally homogeneous pseudo-Riemannian spaces and prove a theorem on their structure. Using three-dimensional Lie groups and the six-dimensional Heisenberg group [11], we construct some examples showing the difference between the Riemannian and pseudo-Riemannian cases for such spaces.     

Unit Tangent Sphere Bundles and Two-Point Homogeneous Spaces

We characterize two-point homogeneous spaces, locally symmetric spaces, C and B-spaces via properties of the standard contact metric structure of their unit tangent sphere bundle. Further, under various conditions on a Riemannian manifold, we show that its unit tangent sphere bundle is a (locally) homogeneous contact metric space if and only if the manifold itself is (locally) isometric to a two-point homogeneous space.     

Homogeneous spaces of curvature bounded below

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces. Dedicated to the memory of A. D. Alexandrov     

KILLING VECTOR FIELDS OF CONSTANT LENGTH ON LOCALLY SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of the circle S ¹. Examples of unit Killing vector fields generated by almost free but not free actions of S ¹ on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated. DOI: .     

Four-dimensional homogeneous Lorentzian manifolds

Four-dimensional locally homogeneous Riemannian manifolds are either locally symmetric or locally isometric to Riemannian Lie groups. We determine how and to what extent this result holds in the Lorentzian case.     

Three-step Harmonic Solvmanifolds

The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with algebraic rank one and two-step nilradical. Our main result shows that the Damek–Ricci spaces are the only harmonic manifolds of this type.     

On a Generalization of Curvature Homogeneous Spaces

Sekigawa proved in 1977 that a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the sense of I.M. Singer is always locally homogeneous. We deal here with the modification of the curvature homogeneity which is said to be “of type (1, 3)”. We give example of a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the modified sense but still not locally homogeneous.     

ON THE INVARIANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLD

In this paper, the vertical and horizontal distributions of an invariant submanifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.     

Affine Versions of Singer's Theorem on Locally Homogeneous Spaces

A theorem of Singer says that an infinitesimaly homogeneous Riemannian manifold is locally homogeneous. We propose two result on affine connections similar to the theorem of Singer. As an application we prove a theorem giving a sufficient condition for local homogeneity in case of affine connections on 2-dimensional manifolds.     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

Natural diagonal Riemannian almost product and para-Hermitian cotangent bundles

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. Studying the compatibility and the anti-compatibility relations between the determined structures and a natural diagonal metric, we find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. Finally, we prove the characterization theorem for the natural diagonal (almost) para-Kählerian structures on the total space of the cotangent bundle.     

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

     

On product affine hyperspheres in R~(n+1)

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space R~(n+1) which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3-and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.     

On two theorems of I. M. Singer about homogenous spaces

A theorem of I. M. Singer [9] states that a Riemannian manifold is locally homogeneous if and only if the Riemannian curvature tensor and its covariant derivatives are the same at each point up to some orderk _M + 1.In the present paper we reprove this theorem by a more direct approach.By using the same approach we also prove, in addition, that a homogeneous Riemannian manifold is completely determined by the curvature and its covariant derivatives at some point up to orderk _M + 2. Moreover, we show how to reconstruct a homogeneous Riemannian manifold only from these curvature data. Finally, we formulate precisely and prove a statement which was announced without proof by Singer in [9].This work was partially supported by the M. P. I. fondi 40%.     

Locally Desarguesian spaces

In Riemannian spaces, locally Desarguesian spaces have constant curvature and are therefore locally symmetric. This does not hold for Finsler spaces, so that locally Desarguesian spaces represent a generalization other than the obvious one we studied previously of (certain) Riemannian symmetric spaces. In this paper we discuss them in detail; as an example of the results obtained we mention that a simply connected locally Desarguesian space without conjugate points is globally Desarguesian. Applications are then given to spaces which are locally symmetric in a wider sense. We also study (and in Minkowski spaces determine exactly) the properties of functions which measure the distance of a point from those on a line.     

Foliations of the Sasakian Space R2n+1 by Minimal Slant Submanifolds

Examples of slant submanifolds in the Sasakian space R²ⁿ⁺¹ are obtained as the leaves of a harmonic, Riemannian 3-dimensional foliation. With the exception of the anti-invariant ones, these leaves are all locally homogeneous manifolds with negative scalar curvature, whose Ricci tensor satisfies (S)(X, X) = 0 for all tangent vector fields.     

Holomorphically projective transformations on complex Riemannian manifold

The aim of the paper is to prove that if a complex Riemannian manifold with holomorphic characteristic connection is holomorphically projective equivalent to a locally symmetric space then it is a complex Riemannian manifold of pointwise constant holomorphic characteristic sectional curvature.Dedicated to N.K. Stephanidis on the occasion of his 65 th birthday.