Some classes of almost contact metric manifolds and contact Riemannian submersions

Locally conformal almost quasi-Sasakian manifolds are related to the Chinea--Gonzales classification of almost contact metric manifolds. It follows that these manifolds set up a wide class of almost contact metric manifolds containing several interesting subclasses. Contact Riemannian submersions whose total space belongs to each of the considered classes are then investigated. The explicit expression of the integrability tensor and of the mean curvature vector field of each fibre are given. This allows us to state the integrability of the horizontal distribution and/or the minimality of the fibres in particular cases. The classes of the base space and of the fibres are also determined, so extending several well-known results.     

On the stochastic geometrical foundations of metric multidimensional scaling

This paper deals with the topological and measure theoretic probability foundations of metric multidimensional scaling. The first four sections develop minimal separation properties for a set of points or space to model a stimulus. The next three sections give measure theoretic probabilistic structure to random stimulus points and their random connecting paths.     

Dirac-Lu space with pseudo-Riemannian metric of constant curvature

In this paper, we will discuss the geometries of the Dirac-Lu space whose boundary is the third conformal space N defined by Dirac and Lu. We firstly give the SO(3, 3) invariant pseudo-Riemannian metric with constant curvature on this space, and then discuss the timelike geodesics. Finally we get a solution of the Yang-Mills equation on it by using the reduction theorem of connections.     

Canonical variation of a Lorentzian metric

Given a Lorentzian manifold (M,_gL)$(M,g_L)$ and a timelike unitary vector field E  , we can construct the Riemannian metric _gR=_gL+2ω⊗ω$g_R=g_L+2\omega \otimes \omega$, ω being the metrically equivalent one form to E. We relate the curvature of both metrics, especially in the case of E   being Killing or closed, and we use the relations obtained to give some results about (M,_gL)$(M,g_L)$.     

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.     

On Einstein extensions of nilpotent metric Lie algebras

The main result of the article is as follows: If a nilpotent noncommutative metric Lie algebra (n, Q) is such that the operator Id ? trace(Ric) / trace(Ric²) Ric is positive definite then every Einstein solvable extension of (n, Q) is standard. We deduce several consequences of this assertion. In particular, we prove that all Einstein solvmanifolds of dimension at most 7 are standard.     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

On the metrizability of cone metric spaces

We have shown in this paper that a (complete) cone metric space (X,E,P,d) is indeed (completely) metrizable for a suitable metric D. Moreover, given any finite number of contractions f₁,…,fn on the cone metric space (X,E,P,d), D can be defined in such a way that these functions become also contractions on (X,D).     

On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings

The purpose of this paper is to establish some theorems on convergence of a sequence in complete metric spaces. As applications, some results of Ghosh and Debnath [J. Math. Anal. Appl. 207 (1997) 96-103], Kirk [Ann. Univ. Mariae Curie-Sk?odowska Sect. A LI.2, 15 (1997) 167-178] and Petryshyn and Williamson [J. Math. Anal. Appl. 43 (1973) 459-497] are obtained from our results as special cases. Also, we give comments on some results in [J. Math. Anal. Appl. 207 (1997) 96-103, J. Math. Anal. Appl. 43 (1973) 459-497]. Some examples are introduced to support our comments.     

The HeatKernel ofthe Sym m etric Space P_n

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｈｅａｔｋｅｒｎｅｌｏｆａｍａｎｉｆｏｒｌｄｃｏｎｔａｉｎｓｒｉｃｈｉｍｆｏｒｍａｔｉｏｎｏｆｇｅｏｍｅｔｒｙａｎｄａｎａｌｙｓｉｓｏｆｔｈｅｍａｎｉｆｏｌｄ．Ｒｅｃｅｎｔｙ，ｍａｎｙｍａｔｈｅｍａｔｉｃｉａｎｓａｎｄｐｈｙｓｉｓｔｓａｒｅｉｎｔｅｒｅｓｔｅｄｉｎｔｈｉｓｆｉｅｌｄａｎｄａｌｏｔｏｆｉｍｐｏｒｔａｎｔｒｅｓｕｌｔｓａｒｅｏｂｔａｉｎｅｄ．ＬｅｔＰｎ（Ｃ）ｄｅｎｏｔｅｔｈｅｓｅｔｏｆａｌｌｎ×ｎｃｏｍｐｌｅｘＨｅｒｍｉｔｅｐｏｓｉｔｉｖｅｄｅｆ…     

Area and coarea formulas for the mappings of Sobolev classes with values in a metric space

We prove the metric differentiability and approximate metric differentiability for some broad classes of mappings with values in a metric space, including Sobolev classes. By way of application, we derive some metric analogs of area and coarea formulas.     

On a problem of iteration invariants for distributional chaos

We show that if f is a DC3 continuous map of a compact metric space then also f^N is DC3, for every N > 0. This solves a problem given by [Li R. A note on the three versions of distributional chaos. Commun Nonlinear Sci Numer Simulat 2011;16:1993-1997].     

Common fixed points of generalized contractions on partial metric spaces and an application

In this paper, common fixed point theorems for four mappings satisfying a generalized nonlinear contraction type condition on partial metric spaces are proved. Presented theorems extend the very recent results of I. Altun, F. Sola and H. Simsek [Generalized contractions on partial metric spaces, Topology and its applications 157 (18) (2010) 2778-2785]. As application, some homotopy results for operators on a set endowed with a partial metric are given.     

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.     

Some conformal and projective scalar invariants of Riemannian manifolds

It is proved that, on any closed oriented Riemannian n-manifold, the vector spaces of conformal Killing, Killing, and closed conformal Killing r-forms, where 1 ≤ r ≤ n ? 1, have finite dimensions t _r , k _r , and p _r , respectively. The numbers t _r are conformal scalar invariants of the manifold, and the numbers k _r and p _r are projective scalar invariants; they are dual in the sense that t _r = t _n?r and k _r = p _n?r . Moreover, an explicit expression for a conformal Killing r-form on a conformally flat Riemannian n-manifold is given.     

On the Manifold of Almost Complex Structures

Let (M, g ₀) be a smooth closed Riemannian manifold of even dimension 2_n admitting an almost complex structure. It is shown that the space of all almost complex structures on M determining the same orientation as the one determined by a fixed almost complex structure J ₀ is a smooth locally trivial fiber bundle over the space of almost complex structures orthogonal with respect to g ₀ and determining the same orientation as J ₀.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 66–71.Original Russian Text Copyright © 2005 by N. A. Daurtseva.     

关于具有常平均曲率和数量曲率超曲面的Moebius几何的一个注记

设x：M→S^n＋1（n≥23）是n＋1-维单位球中的无脐点超曲面，Moebius不变量G，Ф，A和B分别表示x的Moebius度量，Moebius形式，Blaschke形式和Moebius第二基本形式．本文证明了如果x的Moebius形式圣平行，并且A＋λG＋μB=0，其中λ，μ分别是定义在M上的光滑函数，那么Ф=0，由此及李海中、王长平（2003年）文献中的分类定理给出了眇州中具有平行的Moebius形式及满足A＋λG＋μB=0的超曲面的分类．此结果推广了他们及张廷枋（2003年）文献中的结果．     

On the curvature of a generalization of contact metric manifolds

Summary We consider a genaralization of contact metric manifolds given by assignment of 1-formsη¹, . . . ,η^sand a compatible metric gon a manifold. With some integrability conditions they are called almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva"; mso-bidi-font-family:"Monotype Corsiva"'>S-manifolds. We give a sufficient condition regarding the curvature of an almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva";mso-bidi-font-family: "Monotype Corsiva"'>S-manifold to be locally isometric to a product of a Euclidean space and a sphere.