On geodesic mappings of equidistant generalized Riemannian spaces

In this paper geodesic mappings of equidistant generalized Riemannian spaces are discussed. It is proved that each equidistant generalized Riemannian space of basic type admits non-trivial geodesic mapping with preserved equidistant congruence. Especially, there exists non-trivial geodesic mapping of equidistant generalized Riemannian space onto equidistant Riemannian space. An example of geodesic mapping of an equidistant generalized Riemannian spaces is presented.     

On generalized Cartan subspaces

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ and G _k (resp. H _k ) the set of k-rational points of G (resp. H). The variety G _k /H _k is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L ²(G _k /H _k ) of square integrable functions decomposes into several series, one for each H _k -conjugacy class of Cartan subspaces of G _k /H _k .     

On subspaces of pseudoradial spaces

A topological space is pseudoradial if each of its non closed subsets has a sequence (not necessarily with countable length) convergent to outside of . We prove the following results concerning pseudoradial spaces and the spaces , where is an ultrafilter on :

(i) CH implies that, for every ultrafilter on , is a subspace of some regular pseudoradial space.

(ii) There is a model in which, for each P-point , cannot be embedded in a regular pseudoradial space while there is a point such that is a subspace of a zero-dimensional Hausdorff pseudoradial space.

     

Classification of five-dimensional generalized pointwise symmetric Riemannian spaces

Riemannian symmetric spaces have the following two classes of spaces as their natural generalizations: (A) the class ofGS-spaces (generalized symmetric Riemannian spaces); (B) the class ofGPS-spaces (generalized pointwise symmetric Riemannian spaces). A result due to O. Kowalski says that the relation between the two classes is (A) (B), the inclusion being strict. In the present paper the author proves that in dimension 5 the class (A) and the class (B) must coincide. Consequently the explicit classification of five-dimensional GPS-spaces is obtained.     

On generalized recurrent Riemannian manifolds

The object of the present paper is to study a type of Riemannian manifolds called generalized recurrent manifolds. We have constructed two concrete examples of such a manifold whose scalar curvature is non-zero non-constant. Some other properties have been considered. Among others it is shown that on a generalized recurrent manifold with constant scalar curvature, Weyl-semisymmetry and semisymmetry are equivalent. Sufficient condition for a generalized recurrent manifold to be a special quasi Einstein manifold is obtained.     

On complemented subspaces of simplex spaces

Jellett's theorem about the resolution of the space, A(X) of continuous affine functions on a compact ahoquet simplex X into a direct sum is generalized to simplex spaces. A new characteristic property of the space A(X), dual tol ¹(Γ), is given.     

On pseudo-complemented subspaces of Banach spaces

As natural generalizations of complemented subspaces, we introduce pseudo-complemented and strongly pseudo-complemented subspaces of Banach spaces, and we study for such subspaces some of the problems analogous to the classical problems on complemented and quasi-complemented subspaces.     

Variational theory of Riemannian subspaces

If a subspace of a Riemannian manifold is varied the tensors associated with the subspace also vary. The appropriate concept of tensorial variation is introduced and the tensorial variations of several tensors are calculated. A general formula for the variation of an integral is given. Applications are made to the mean curvature integrals of hypersurfaces.     

On subspaces of Banach spaces without quasicomplements

It is proved that for uncountableΓ,c ₀(Γ) has no quasicomplement inm(Γ). Research supported by the National Science Foundation, U.S.A.     

On finite dimensional subspaces of Banach spaces

The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces. The contribution to the paper of the second author is part of his Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor A. Dvoretzky. The authors wish to thank Professor A. Dvoretzky for the interest he showed in the paper and for his helpful advice.     

On the geodesic mobility of Riemannian spaces

It is proved that the degree of geodesic mobility of ann-dimensional Riemannian space (M, g) which is not a space of constant curvature can take only the valuesp=m(m+1)/2+l, wherem is the number of linearly independent concircular covector fields onM andl ranges from 1 to [(n+1−m)/3]; the brackets denote the integer part of a number. Thus the problem of finding all lacunas in the distribution of degrees of geodesic mobility is completely solved for this class of spaces. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp 620–626, October, 2000.     

On the structure of geodesic orbit Riemannian spaces

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterized by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial principal isotropy algebras. We discuss also some new examples of geodesic orbit Riemannian spaces, new methods to obtain such examples, and some unsolved questions.     

On three-dimensional Riemannian spaces with curvature bounded above

For three-dimensional closed simply-connected Riemannian spaces one gets a lower bound for the radius of injectivity of the exponential map from a lower bound of the sectional and an upper bound of the Ricci curvatures.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 881–887, June, 1973.     

On generalized symmetric Finsler spaces

In this paper, we study generalized symmetric Finsler spaces. We first study some existence theorems, then we consider their geometric properties and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each generalized symmetric Finsler space is of finite order and those of even order reduce to symmetric Finsler spaces and hence are Berwaldian.