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.     

Conformal mappings onto Einstein spaces

We study conformal mappings of Riemannian manifolds onto Einstein manifolds under minimal condition on the differentiability class of manifolds in question. We establish under what conditions the linear equations obtained by J. Mike?, M. L. Gavril’chenko and E. I. Gladyscheva that define such mappings.     

Closed mappings onto metric spaces

We investigate the classes of spaces that can be mapped onto a metrizable space by a closed mapping with fibers having a given property $\text{P}$. We give some conditions which assure that such classes are closed under the action of perfect or open and compact mappings. Such a treatment includes the investigation of paracompact p-spaces and M-spaces. We also discuss spaces that can be mapped onto a metacompact Moore space.     

Decomposable symmetric mappings between infinite-dimensional spaces

Decomposable mappings from the space of symmetric k-fold tensors over E, , to the space of k-fold tensors over F, , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.     

Almost-geodesic mappings of affinely connected and Riemann spaces

Tensor methods are used to present the fundamental facts of the theory of geodesic mappings of affinely connected spaces with and without torsion.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 13, pp. 3–26, 1982.     

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.     

Totally geodesic submanifolds of maximal rank in symmetric spaces

We gave a complete list of totally geodesic submanifolds of maximal rank in symmetric spaces of noncompact type. The compact cases can be obtained by the duality.     

Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces

We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of a result on the asymptotic behavior of Picard iterates for firmly nonexpansive mappings proved by Reich and Shafrir. From this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting.     

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Dedicated to professor K. Oikawa on his 60th birthday     

The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S ₃ onto a genus 2 Riemann surface S ₂ by Hol(S ₃, S ₂). Call two mappings f and g in Hol(S ₃, S ₂) equivalent whenever there exist conformal automorphisms α and β of S ₃ and S ₂ respectively with f ? α = β ? g. It is known that Hol(S ₃, S ₂) always consists of at most two equivalence classes.We obtain the following results: If Hol(S ₃, S ₂) consists of two equivalence classes then both S ₃ and S ₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S ₃, S ₂) there exist anticonformal automorphisms α? and β? with f ? α? = β? ? g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S ₃, S ₂) such that Hol(S ₃, S ₂) consists of two equivalence classes.     

Totally geodesic submanifolds in compact Riemannian symmetric spaces and their stability

In this paper, we determine a large class of totally geodesic submanifolds of a compact Riemannian symmetric space. The stability of these submanifolds in their ambient space is also determined.     

On canonical almost geodesic mappings of the first type of affinely connected spaces

In this paper, we study special cases of canonical almost geodesic mappings of the first type of affinely connected space?The basic equations of mappings in question are reduced to a closed system of Cauchy type in covariant derivatives, and the number of essential parameters in the general solution of this system is estimated. We give an example of such mappings from a flat space onto another flat space. The mappings constructed send straight lines of the first space into parabolas in the second space. These almost geodesic mappings of the first type do not belong to the classes of mappings of the second and third types.     

The mobility of Riemannian spaces with respect to conformal mappings onto Einstein spaces

We give an estimate of the first lacuna in the distribution of mobility degrees r of n-dimensional (pseudo-)Riemannian spaces with respect to conformal mappings onto Einstein spaces. We obtain a tensor characteristic of spaces which are not conformally flat and have r = n − 1, which is the maximum possible value. Thus, we have found maximum mobile nonconformally flat spaces with r = n − 1.     

Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

It is established that if and are connected locally symmetric spaces of noncompact type where has finite volume, and is a totally geodesic immersion, then the closure of in is an immersed ``algebraic' submanifold. It is also shown that if in addition, the real ranks of and are equal, then the the closure of in is a totally geodesic submanifold of The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.