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Special vector fields, such as conformal vector fields and Killing vector fields, are commonly used in studying the geometry of a Riemannian manifold. Though there are Riemannian manifolds, which do not admit certain conformal vector fields or certain Killing vector fields, respectively. Closed vector fields exist in abundance on each Riemannian manifold. In this paper, we used closed vector fields to study the geometry of the Riemannian manifold. In the first result, we showed that a compact Riemannian manifold $ (M^{n}, g) $ admits a closed vector field $\boldsymbol{\omega }$ with $ \mathrm{div} \boldsymbol{\omega }$ non-constant and an eigenvector of the rough Laplace operator, the integral of the Ricci curvature $ Ric(\boldsymbol{\omega }, \boldsymbol{\omega }) $ has a suitable lower bound that is necessarily isometric to $ S^{n}(c) $ and that the converse holds. In the other result, we found a characterization of an Euclidean space using a closed vector field $\boldsymbol{\omega }$ with non-constant length that annihilates the rough Laplace operator and squared length of its covariant derivative that has a suitable upper bound. Finally, we used the closed vector field provided by the gradient of the non-trivial solution of the Fischer-Marsden equation on a complete and simply connected Riemannian manifold $ (M, g) $ and showed that it is necessary and sufficient for $ (M, g) $ to be isometric to a sphere and that the squared length of the covariant derivative of this closed vector field has a suitable upper bound.

A concircular vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ induces a vector field $ \mathbf{w} $ on an orientable hypersurface $ M $ of the unit sphere $ \mathbf{S}^{n+1} $, simply called the induced vector field on the hypersurface $ M $. Moreover, there are two smooth functions, $ f $ and $ \sigma $, defined on the hypersurface $ M $, where $ f $ is the restriction of the potential function $ \overline{f} $ of the concircural vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ to $ M $ and $ \sigma $ is defined as $ g\left(\mathbf{u}, N\right) $, where $ N $ is the unit normal to the hypersurface. In this paper, we show that if function $ f $ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $ \mathbf{w} $ has a certain lower bound, then a characterization of a small sphere in the unit sphere $ \mathbf{S}^{n+1} $ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $ M $ in the direction of the vector field $ \mathbf{w} $ with a non-zero function $ \sigma $.

Abstract We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by u = − φ ( N ) $\mathbf{u}=-\varphi (N)$ . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere S 2 n + 1 $\mathbf{S}^{2n+1}$ . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature R i c ( u , u ) $Ric ( \mathbf{u},\mathbf{u} ) $ of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is S 2 n + 1 $\mathbf{S}^{2n+1}$ .

The first results of the physical modeling of positive multistrike lightning formation processes using positively charged artificial thunderstorm cells are presented. Experimental studies have shown a significant influence of the number of thunderstorm cells and groups of model hydrometeors introduced into them on the probability of the initiation of repeated strikes. It was found that with an increased number of cells and groups of model hydrometeors, the probability of the formation of repeated positive discharges increases several times. When the second group of model hydrometeors has been introduced into the artificial thunderstorm cell, or when the number of cells has been increased, the probability of the repeated discharge initiation has increased almost in three and in four times respectively. It has been revealed that, depending on the arrangement of model hydrometeor groups in artificial thundercloud cells, the formation of repeated positive discharges from them may proceed both with the “connection” of the uncharged areas of the lower cell and with the “connection” of the upper cell. The parameters of the first and repeated impulse current pulses between the positively charged cells and the ground were determined. It was found that with an increasing number of model hydrometeor groups, the value of the charge neutralized during the stages of first and repeated discharge formation increases. When forming multistrike positive discharges with the “connection” of the upper artificial thunderstorm cell, 20–30% more cloud charge has been neutralized during the repeated discharge than during the formation of a single positive discharge. It was found that the formation of positive repeated discharges was observed in about half of the cases, and that the radiation power and impulse current amplitude at repeated discharges are higher than at the first discharge. This article discusses some possible reasons for such a ratio between the parameters of the first and repeated discharges. It is assumed that the discovered significant influence of large model hydrometeor groups on the probability of the formation and the characteristics of repeated positive discharges from artificial thunderstorm cells indicate a possible key influence of hail arrays in the thundercloud on the formation of repeated strikes of positive lightning and bipolar lightning. The obtained results show that artificial thunderstorm cells of positive polarity, together with groups of large model hydrometeors, have prospects for physical modeling and the investigation of processes of the formation of positive and bipolar repeated lightning strikes.

The results of the physical modelling of the formation processes of upward discharges from grounded objects in the artificial thunderstorm cell’s electric field are presented. We established the considerable influence of the electrode tip’s radius on the pulse streamer corona stem’s parameters and, subsequently, on the probability of the transformation of the impulse streamer corona first flash’s stem into a first stage of upward leader. We determined the diapason of the optimal tip radii for a lightning rod or lightning conductor, which allows for the most probable formation of the first impulse streamer corona, with the parameters providing the best conditions for the upward leader’s start, the purpose of which is the lowering of the probability of lightning striking the object under protection. A considerable difference between the electrical characteristics of the first impulse corona flash with and without the streamer–leader transition was established. It was shown that the amplitude of the streamer corona flash current impulse is considerable, but not the main defining factor of the streamer–leader transition. It is established that the charge value of the streamer corona first flash is not a threshold requirement for the formation of the upward leader from a ground object, but only defines the probability of the successful upward leader formation. Based on the analysis of the experimental data received, we suggest that there is a dependency between the probability of upward positive leader formation from the grounded objects and the charge value of the first pulse streamer corona flash for the rod (centered) and rope (elongated) lightning conductors and objects in the electric field of the thundercloud and downward lightning leader. The obtained results can be used for mathematical modelling of the formation processes of upward discharges from grounded objects in the artificial thunderstorm’s electric field, as in a natural thunderstorm situation.

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold Mn of Sasakian space forms M2m+1ε. As Chen–Ricci inequality applications, we found the characterization of the base of the warped product Mn via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere Sp.

Au début des années 1990, le discours des réformateurs libéraux comporte une vision très négative de la révolution russe d’Octobre 1917, présentée comme une véritable « catastrophe ». L’ensemble de l’expérience soviétique est d’ailleurs interprété comme une succession de mauvais choix qui ont contribué à la consolidation d’un système politique et économique inefficace. Une telle vision s’inscrit parfaitement dans l’argumentation des réformateurs qui l’instrumentalisent pour légitimer leurs choix politiques. Toutes les décisions sont justifiées par la nécessité de rompre définitivement avec l’ancien système communiste. Dans le contexte de stabilisation autoritaire des années 2000, la radicalité des transformations que le pays a connues à la fin du XXe siècle est de plus en plus critiquée. La ressemblance entre le caractère utopique du projet bolchevique et du projet libéral ainsi que l’utilisation d’une logique de légitimation comparable est mise en avant, ce qui pousse les représentants du courant libéral russe à repenser leur expérience et à s’interroger sur le caractère révolutionnaire des réformes du début des années 1990. Le présent article explore l’évolution de l’usage, dans le camp libéral, du paradigme « révolution » pour désigner les réformes post-soviétiques, allant de son rejet initial à son acceptation dans une perspective comparative, puis à une remise en question de sa pertinence.

The results of experimental laboratory investigations of possible mechanisms of the impact of lightning and thunderclouds on aircraft radomes and equipment inside them are presented. An artificial thunderstorm cell of negative polarity and model aircraft radomes with lightning diverter strips have been used. Experiments have shown that the discharge processes in a radome model significantly depend on the magnitude of the charge that accumulates on the inner and outer surfaces of the radome shell. It is established that the accumulation of large-magnitude charges of different signs on the outer and/or inner surface of the radome (up to hundreds of µC/m2) shell leads to a multivariance of the mechanisms of development of discharge processes inside the radome model, along its surface, and in the space near it. Significant influence of the “reverse” discharge from the antenna model under the radome on the types of current impulses recorded on the antennas under impact of the artificial thunderstorm cell is established. Peculiarities of the discharge formation in the radome model when using solid and segmented diverter strips for its protection are revealed. Parameters of the current impulses registered on the diverter strips and the antennas have been determined. Based on the conducted research, the possible mechanisms of the impact of thunderclouds and lightning discharges on radio-transparent aircraft radomes and the equipment inside them are considered.

The article represents results of a physical simulation of incomplete upward leader discharges induced on air transmission lines’ elements, using charged artificial thunderstorm cells of negative polarity. The influence of such discharges on closely located model sensors (both of rod and elongated types) of digital monitoring systems, as well as on the models of receiver-transmission systems of local data collection (antennas), was determined. Effect of heterogeneity of electromagnetic field caused by incomplete upward discharges on frequency specter of signals generated on sensors and antennas was estimated. Wavelet analysis was carried out to determine the basic frequency diapasons of such signals. Based on experimental data obtained, suppositions about the extent of influence of nearby incomplete leader discharges on the functioning of currently used systems of transmission lines’ monitoring were made.

The results of a physical simulation using negatively charged artificial thunderstorm cells to test the spectrum of possible electromagnetic effects of upward streamer discharges on the model elements of transmission line monitoring systems (sensor or antennas) are presented. Rod and elongated model elements with different electric field amplification coefficients are investigated. A generalization is made about the parameters of upward streamer current impulse and its electromagnetic effect on both kinds of model elements. A wavelet analysis of the upward streamer corona current impulse and of the signal simultaneously induced in the neighboring model element is conducted. A generalization of the spectral characteristics of the upward streamer current and of the signals induced by the electromagnetic radiation of the nearby impulse streamer corona on model elements is made. The reasons for super-high and ultra-high frequency ranges in the wavelet spectrum of the induced electromagnetic effect are discussed. The characteristic spectral ranges of the possible electromagnetic effect of upward streamer flash on the elements of transmission line monitoring systems are considered.

In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.

Our purpose is to study a space Π of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space Π are set in the generalized fiberings. Fields of these affine connection objects define torsion and curvature tensors. The canonical cases of planar and normal generalized affine connections are considered.

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold ( M , g ) with potential function f, there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇ f . We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere S n ( c ) and the Euclidean space E n . We also show that the connecting function influences on a topology of the Riemannian manifold.

The space Π of centered m-planes is considered in projective space P n . A principal bundle is associated with the space Π and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space Π . Semi-normalized spaces Π 1 , Π 2 and normalized space Π 1 , 2 are investigated. By virtue of the Cartan−Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space Π to the normalized space Π 1 , 2 .

The article is devoted to the comparative analyses of mythological motifs spread in the folk Slavic and Jewish traditions and connected with the beliefs about the bird-like demons. The authors examine the legends and superstitions from the regions of close ethniccultural contacts, such as Western Ukraine, Western Belarus, and Poland, where the Slavic tradition had long-lasting interconnection with non-Slavic cultures. The aim of the investigation is to find parallels with the Slavic folklore evidences of bird features, which are attributed to some supernatural creatures in folk culture. In particular folk beliefs about the bird-legged demons, which can be identified by their footprints, may have their roots in the Talmud tradition. We can suppose that these motifs came to the Old Russia literature (treatises against the pagans) through the Byzantine mediation. As for the Ukrainian and Byelorussian folk beliefs about the bird-like demons, some of them could have been borrowed from the folklore of their neighbors – the Jews. This investigation was made in the framework of the research project “The Slavic Mythology: Results and Perspectives of Interdisciplinary Studies” supported by the Russian Foundation of Basic Research (grant 06-06-80095).