Recent zbMATH articles in MSC 53https://zbmath.org/atom/cc/532021-01-08T12:24:00+00:00WerkzeugOn classification of factorable surfaces in Galilean 3-space \(\mathbb{G}^3\).https://zbmath.org/1449.530042021-01-08T12:24:00+00:00"Bansal, Pooja"https://zbmath.org/authors/?q=ai:bansal.pooja"Shahid, Mohammad Hasan"https://zbmath.org/authors/?q=ai:shahid.mohammed-hasanSummary: In this paper, we study factorable surfaces in Galilean 3-space \(\mathbb{G}^3\).
Then we describe, up to a congruence, factorable surfaces and the several results in this respect are obtained. In particular, factorable surfaces in terms of an isometric immersion, finite type Gauss map and the pointwise 1-type Gauss map of the surfaces are considered and the characterization results on the factorable surfaces with respect to these conditions are obtained.Einstein-Weyl structures on almost cosymplectic manifolds.https://zbmath.org/1449.530492021-01-08T12:24:00+00:00"Chen, Xiaomin"https://zbmath.org/authors/?q=ai:chen.xiaominIf \(M\) is a \((2n+1)\)-dimensional smooth manifold, then an almost contact structure on \(M\) is a triple \((\varphi,\xi,\eta)\), where \(\varphi\) is a \((1,1)\)-tensor field, \(\xi\) is a unit vector field, called Reeb vector field, \(\eta\) is a one-form dual to \(\xi\) satisfying \(\varphi^2=-I+\eta\otimes\xi\) and \(\eta\circ\varphi=0\). A smooth manifold with such a structure is called an almost contact manifold. A Riemannian metric \(g\) on \(M\) is called compatible with the almost contact structure if \(g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)\), \(g(X,\xi)=\eta(X)\) for any vector fields \(X\), \(Y\). An almost contact structure together with a compatible metric \(\Sigma=(\varphi,\xi,\eta,g)\) is called an almost contact metric structure and \((M,\Sigma)\) is called an almost contact metric manifold. An almost contact structure \((\varphi,\xi,\eta)\) is said to be normal if the corresponding complex structure \(J\) on \(M\times\mathbb{R}\) is integrable. If \(\omega\) is the fundamental 2-form on \(M\) defined by \(\omega(X,Y)=g(\varphi X,Y)\), then an almost contact metric manifold \((M,\Sigma)\) such that the fundamental form \(\omega\) and 1-form \(\eta\) satisfy \(\eta=0\) and \(d\omega=2\alpha\,\eta\wedge\omega\) is called an almost \(\alpha\)-cosymplectic manifold, where \(\alpha\) is a real number. A normal almost \(\alpha\)-cosymplectic manifold is called \(\alpha\)-cosymplectic manifold. \(M\) is an almost cosymplectic manifold if \(\alpha=0\) and is called an almost Kenmotsu manifold if \(\alpha=1\). An almost cosymplectic manifold \((M,\Sigma)\) is called a \(K\)-cosymplectic manifold if the Reeb vector field \(\xi\) is Killing. If the curvature tensor \(R\) of an almost cosymplectic manifold \((M,\Sigma)\) satisfies \(R(X,Y)g=\kappa(\eta(Y)X-\eta(X)Y)+\mu(\eta(Y)hX-\eta(X)hY)\) for any vector fields \(X\), \(Y\), where \(\kappa\), \(\mu\) are constant and \(h=\frac12\mathcal{L}_\xi\varphi\), then \(M\) is called an almost cosymplectic \((\kappa,\mu)\)-manifold. If \((M,c=[g])\) is a conformal manifold with conformal class \(c\), then a torsion-free linear connection \(D\) is called a Weyl connection if it preserves the conformal class \(c\). For any metric \(g\) in \(c\) it carries a 1-form \(\theta\), called the Lee form with respect to \(g\), such that \(Dg=-2\,\theta\times g\). It is related to the Levi-Civita connection \(\nabla\) by the following relation: \(D_XY=\nabla_XY+\theta(Y)X-g(X,Y)B\), where \(B\) is dual to \(\theta\) with respect to \(g\). A Weyl structure \(W=(D,[g])\) is said to be closed, or exact if its Lee form is closed, or exact, respectively, with respect to any metric in \(c\). A Weyl structure \(W=(D,[g])\) is called Einstein-Weyl if the trace-free component of the symmetric part of \(\mathbf{Ric}^D\) is identically zero, namely there exists a smooth function \(\Lambda\) such that \(\mathbf{Ric}^D(Y,X)+\mathbf{Ric}^D(X,Y)=\Lambda g(Y,X)\).
In this paper, the author studies Einstein-Weyl structures on almost cosymplectic manifolds. First, the author proves that if a \((2n+1)\)-dimensional almost \((\kappa,\mu)\)-cosymplectic manifold \(M\) admits a closed Einstein-Weyl structure is an Einstein manifold or a cosymplectic manifold, and if \(M\) admits two Einstein-Weyl structures with \(\theta\) and \(-\theta\) is either cosymplectic or Einstein. Next, it is shown that if a \(3\)-dimensional compact almost \(\alpha\)-cosymplectic manifold \((M,\varphi,\xi,\eta,g)\) that admits a closed Einstein-Weyl structure, then \(M\) is Ricci-flat. Also, if a \((2n+1)\)-dimensional almost \(\alpha\)-cosymplectic manifold \((M,\varphi,\xi,\eta,g)\) admits two Einstein-Weyl structures with \(\pm\theta\) and the Ricci tensor \(Q\) is commuting, i.e., \(\varphi Q=Q\varphi\), then \(M\) is either an Einstein manifold, or an \(\alpha\)-cosymplectic manifold. Finally, the author shows that if a \((2n+1)\)-dimensional compact \(K\)-cosymplectic manifold \((M,\varphi,\xi,\eta,g)\) admits a closed Einstein-Weyl structure, then \(M\) is cosymplectic, and if \(M\) admits two Einstein-Weyl structures with \(\pm\theta\), then either \(M\) is Ricci-flat, or the scalar curvature is non-positive and invariant along the Reeb vector field \(\xi\).
Reviewer: Andrew Bucki (Edmond)Invariants of dual surfaces generated by spacelike curves in de Sitter space.https://zbmath.org/1449.530092021-01-08T12:24:00+00:00"Liu, Haiming"https://zbmath.org/authors/?q=ai:liu.haiming"Miao, Jiajing"https://zbmath.org/authors/?q=ai:miao.jiajingSummary: In this paper, we investigate the geometric properties of invariants of the first lightcone dual surfaces and hyperbolic dual surfaces generated by spacelike curves in three dimensional de Sitter space from the view point of contact geometry.On a class of Spray of weakly projective Ricci curvature and its Finsler metrizability.https://zbmath.org/1449.530482021-01-08T12:24:00+00:00"Cheng, Xinyue"https://zbmath.org/authors/?q=ai:cheng.xinyue"Gong, Yannian"https://zbmath.org/authors/?q=ai:gong.yannian"Li, Ming"https://zbmath.org/authors/?q=ai:li.ming.8|li.ming.7|li.ming.4|li.ming.2|li.ming.1|li.ming.9|li.ming.5|li.ming.6|li.ming|li.ming.3Summary: It is important to study the curvature properties and Finsler metrizability of a Spray in Spray geometry. It makes sense to study a kind of projective flat Spray \({\tilde G}\) constructed by Funk metric \(\Theta\), which satisfies \({{\tilde G}^i} = \tau \Theta {y^i}\) and \(\tau\) is a constant. In this paper, we firstly calculate the projective Ricci curvature of \({\tilde G}\). Furthermore, the Finsler metrizability of \({\tilde G}\) is studied under certain conditions of projective Ricci curvature. On the one hand, under the condition that \({\tilde G}\) is projective Ricci flat, the volume form of the manifold can be determined. On the other hand, suppose that \({\tilde G}\) can be induced by Finsler metric \({\tilde F}\) with weak projective Ricci curvature and not projective Ricci flat, then the structure of \({\tilde F}\) can be determined. Finally, the Finsler metric \({\tilde F}\) with weak projective Ricci curvature is preliminarily classified.On a three-dimensional Riemannian manifold with an additional structure.https://zbmath.org/1449.530122021-01-08T12:24:00+00:00"Dzhelepov, Georgi"https://zbmath.org/authors/?q=ai:dzhelepov.georgi-d"Dokuzova, Iva"https://zbmath.org/authors/?q=ai:dokuzova.iva"Razpopov, Dimitar"https://zbmath.org/authors/?q=ai:razpopov.dimitarSummary: We consider a 3-dimensional Riemannian manifold \(M\) with a metric tensor \(g\), and affinors \(q\) and \(S\). We note that the local coordinates of these three tensors are circulant matrices. We have that the third degree of \(q\) is the identity and \(q\) is compatible with \(g\). We discuss the sectional curvatures in case when \(q\) is parallel with respect to the connection of \(g\).Chebyshevian compositions in four dimensional space with an affine connectedness without a torsion.https://zbmath.org/1449.530102021-01-08T12:24:00+00:00"Ajeti, Musa"https://zbmath.org/authors/?q=ai:ajeti.musaSummary: Let \(A_4\) be an affinely connected space without a torsion. Following [\textit{G. Zlatanov} and \textit {B. Tsareva}, ``Conjugated compositions in even-dimensional affinely connected spaces without a torsion'', REMIA 2010, 10--12 December, Plovdiv, Bulgaria, 225--231 (2010)], we define the affinors \(a^\beta_\alpha\) and \(b^\beta_\alpha\), that define conjugate compositions \(X\times\overline X_2\) and \(Y\times \overline Y_2\) in \(A_4\). We define a third composition \(Z\times\overline Z_2\) with the help of the affinor \(\widetilde{c}^\beta_\alpha= ic^\beta_\alpha\), \((i^2=-1)\), where \(c^\beta_\alpha=-a^\beta_\alpha b^\sigma_\alpha\). We have found a necessary and sufficient condition for any of the above composition to be a (ch-ch) composition. We have found the spaces \(A_4\) that contain such compositions. We have shown that if the compositions \(X\times\overline X_2\), \(Y\times\overline Y_2\) and \(Z \times\overline Z_2\) are of the kind (ch-ch) then the space \(A_4\) is affine.The tensor splitting methods for solving tensor absolute value equation.https://zbmath.org/1449.150592021-01-08T12:24:00+00:00"Bu, Fan"https://zbmath.org/authors/?q=ai:bu.fan"Ma, Chang-Feng"https://zbmath.org/authors/?q=ai:ma.changfengSummary: Recently, \textit{W. Li} et al. [Appl. Numer. Math. 134, 105--121 (2018; Zbl 1432.65037)] presented the tensor splitting methods for solving multilinear systems and \textit{S. Du} et al. [Sci. China, Math. 61, No. 9, 1695--1710 (2018; Zbl 1401.15024)] generalized tensor absolute value equations. In this paper, we verify the existence of solutions of tensor absolute value equations and propose the tensor splitting methods for solving this class of equation. Furthermore, the convergence analysis of the tensor splitting method is also studied under suitable conditions. Finally, numerical examples show that our algorithm is an efficient iterative method.Split quaternions and time-like constant slope surfaces in Minkowski 3-space.https://zbmath.org/1449.140102021-01-08T12:24:00+00:00"Babaarslan, Murat"https://zbmath.org/authors/?q=ai:babaarslan.murat"Yayli, Yusuf"https://zbmath.org/authors/?q=ai:yayli.yusufSummary: In the present paper, we prove that time-like constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit time-like split quaternions and also homothetic motions. Afterwards we give some examples to illustrate our main results by using Mathematica.Four-dimensional shrinking gradient Ricci solitons with half positive isotropy curvature.https://zbmath.org/1449.530452021-01-08T12:24:00+00:00"Zhang, Zhuhong"https://zbmath.org/authors/?q=ai:zhang.zhuhongSummary: In this paper, we study four-dimensional shrinking gradient Ricci solitons with half positive isotropy curvature (half-PIC). We show that, the bound of the traceless Ricci curvature Ric will control the bound of the self-dual part of the Weyl tensor \({W_+}\) or the antiself-dual part \(W_-\). In particular, we give a new and simpler proof of the following theorem: Any oriented four-dimensional Einstein manifold with half-PIC must be half conformally flat, and therefore isometric to \({S^4}\) or \(CP^2\) with standard metric. A more general result on shrinking gradient Ricci solitons was given.Inequalities on generalized normalized \(\delta\)-Casorati curvatures for submanifolds in statistical manifolds of constant curvatures.https://zbmath.org/1449.530132021-01-08T12:24:00+00:00"Cai, Dandan"https://zbmath.org/authors/?q=ai:cai.dandan"Liu, Xudong"https://zbmath.org/authors/?q=ai:liu.xudong"Zhang, Liang"https://zbmath.org/authors/?q=ai:zhang.liang.1Summary: We considered submanifolds in statistical manifolds of constant curvatures by using Oprea's optimization method, and obtained some geometric inequalities involving the generalized normalized \(\delta\)-Casorati curvatures. We gave the upper bound and the lower bound of the normalized scalar curvature of the submanifolds, respectively, and the properties of submanifolds satisfying the equality cases.Gromov hyperbolicity of the Kobayashi metric.https://zbmath.org/1449.320092021-01-08T12:24:00+00:00"Andreev, Lyubomir"https://zbmath.org/authors/?q=ai:andreev.lyubomir"Nikolov, Nikolai"https://zbmath.org/authors/?q=ai:nikolov.nikolai-marinov"Trybula, Maria"https://zbmath.org/authors/?q=ai:trybula.mariaThis paper deals with the Kobayashi distance \( k_{\Omega} \) on domains \( \Omega \) in \( \mathbb{C}^{n} \). The authors propose a collection of many known results on the subject. More specially, suppose that \( \Omega \)is bounded convex domain with \( C^{\infty} \) boundary. Then \( (\Omega, k_{\Omega}) \) is Gromov hyperbolic if and only if \( \partial \Omega \) has finite type in the sense of D'Angelo. A special attention is paid on the cases when Gromov hyperbolicity is violated. For example, let \( \Omega \subset \mathbb{C}^{n} \) be a bounded \( \mathbb{C} \) convex domain and \( S \) is a complex affine hyperplane such that \( \Omega \cap S \neq \emptyset \). Then \( (\Omega \setminus S, k_{\Omega \setminus S})\) is not Gromov hyperbolic.
Reviewer: Petar Popivanov (Sofia)Certain results on Kenmotsu pseudo-metric manifolds.https://zbmath.org/1449.530262021-01-08T12:24:00+00:00"Naik, Devaraja Mallesha"https://zbmath.org/authors/?q=ai:naik.devaraja-mallesha"Venkatesha"https://zbmath.org/authors/?q=ai:venkatesha.h|venkatesha.venkatesha"Prakasha, D. G."https://zbmath.org/authors/?q=ai:prakasha.doddabhadrappla-gowdaSummary: In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant \(\varphi\)-sectional curvature, and prove the structure theorem for \(\xi\)-conformally flat and \(\varphi\)-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an \(\eta\)-Einstein Kenmotsu pseudo-metric manifold of dimension higher than 3 admitting a Ricci soliton is Einstein, and a Kenmotsu pseudo-metric 3-manifold admitting a Ricci soliton is of constant curvature \(-\varepsilon\).A note on the reversibility of Finsler manifolds.https://zbmath.org/1449.530192021-01-08T12:24:00+00:00"Yin, Songting"https://zbmath.org/authors/?q=ai:yin.songtingSummary: For a Finsler manifold with the weighted Ricci curvature bounded from below, we give Cheng type and Mckean type comparison theorems for the first eigenvalue of Finsler Laplacian. When the weighted Ricci curvature is nonnegative, we also obtain Calabi-Yau type volume growth theorem. These generalize and improve some recent literatures. Especially, by using the relationship of the counterparts between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility.Slant immersions in \(C_5\)-manifolds.https://zbmath.org/1449.530422021-01-08T12:24:00+00:00"de Candia, Salvatore"https://zbmath.org/authors/?q=ai:de-candia.salvatore"Falcitelli, Maria"https://zbmath.org/authors/?q=ai:falcitelli.mariaSummary: Odd-dimensional non anti-invariant slant submanifolds of an \(\alpha\)-Kenmotsu manifold are studied. We relate slant immersions into a Kähler manifold with suitable slant submanifolds of an \(\alpha\)-Kenmotsu manifold. More generally, in the framework of Chinea-Gonzalez, we specify the type of the almost contact metric structure induced on a slant submanifold, then stating a local classification theorem. The case of austere immersions is discussed. This helps in proving a reduction theorem of the codimension. Finally, slant submanifolds which are generalized Sasakian space-forms are described.\(*\)-Ricci tensors for real hypersurfaces in complex hyperbolic two-plane Grassmannians.https://zbmath.org/1449.530082021-01-08T12:24:00+00:00"Liao, Chunyan"https://zbmath.org/authors/?q=ai:liao.chunyan"Chen, Xiaomin"https://zbmath.org/authors/?q=ai:chen.xiaominSummary: This paper introduces the \(*\)-Ricci tensor from the curvature tensor of a real hypersurface in complex hyperbolic two-plane Grassmannian \(S{U_{2,m}}/S ({U_2}{U_m})\), \(m \ge 2\). It is proved that there are not \(*\)-Einstein metrics on Hopf hypersurfaces of \(S{U_{2,m}}/S ({U_2}{U_m})\). As a generalization of the \(*\)-Einstein metric, we introduce the \(*\)-Ricci soliton and prove that a real hypersurface with a \(*\)-Ricci soliton whose potential vector field is the Reeb vector field, is an open part of a tube around some totally geodesic \(S{U_{2,m-1}}/S ({U_2}{U_{m-1}})\) in \(S{U_{2,m}}/S ({U_2}{U_m})\) or a horosphere whose center at infinity is singular. Finally we study a Hopf hypersurface with pseudo anti-commuting \(*\)-Ricci tensor.Natural connections with torsion expressed by the metric tensors on almost contact manifolds with B-metric.https://zbmath.org/1449.530502021-01-08T12:24:00+00:00"Manev, Mancho"https://zbmath.org/authors/?q=ai:manev.mancho"Ivanova, Miroslava"https://zbmath.org/authors/?q=ai:ivanova.miroslavaSummary: On a main class of the almost contact manifolds with B-metric, it is described the family of the linear connections preserving the manifold's structures by 4 parameters. In this family there are determined the canonical-type connection and the connection with zero parameters.The \(\alpha\)-dual differential geometry of symmetric nonsingular matrices.https://zbmath.org/1449.530212021-01-08T12:24:00+00:00"Wu, Liping"https://zbmath.org/authors/?q=ai:wu.liping"Zhang, Shaoxiang"https://zbmath.org/authors/?q=ai:zhang.shaoxiangSummary: The manifold \(S (n)\) is explored. On the set of all symmetric nonsingular matrices a Riemannian metric is defined and dual \(\alpha\)-connections are obtained. Furthermore, the \(\alpha\)-dual geometric structure is obtained. Finally, an example is given to illustrate our results.Study on geometric evolution properties of planar closed curve flow.https://zbmath.org/1449.530522021-01-08T12:24:00+00:00"Ding, Danping"https://zbmath.org/authors/?q=ai:ding.danping"Cheng, Yongting"https://zbmath.org/authors/?q=ai:cheng.yongtingSummary: In this paper, the governing equation of curve geometric variables is used to discuss the geometric characteristics of plane curve flow, and the description and characterization of the evolution properties of correlation variables are obtained. The global evolution law and characteristics of planar closed curve flow are characterized by the distance from the outer point of the curve to the curve, and the global evolution speed of plane simple closed curve is found to be limited.Spherically-symmetric non-linear sigma model: the exact solutions obtained with isometrical embedding method.https://zbmath.org/1449.830092021-01-08T12:24:00+00:00"Chervon, Sergeĭ Viktorovich"https://zbmath.org/authors/?q=ai:chervon.sergei-viktorovich"Svistunova, Yuliya Aleksandrovna"https://zbmath.org/authors/?q=ai:svistunova.yuliya-aleksandrovnaSummary: The method to generate exact cosmological solutions in the frame of the spherically-symmetric non-linear sigma model offered in the present paper. This method is based on the isometrical embeddings of the target space (non-linear sigma-model fields chiral space) into space-time. Also the method application to two- and three-component chiral spaces embedded into space-time was considered. The exact cosmological solutions were obtained in the frame of the several special cases of the two- and three-component spherical-symmetric non-linear sigma-model. The obtained cosmological solutions were also investigated.The infinitesimal counterpart of tangent presymplectic groupoids of higher order.https://zbmath.org/1449.530252021-01-08T12:24:00+00:00"Kouotchop Wamba, P. M."https://zbmath.org/authors/?q=ai:kouotchop-wamba.p-m"Mba, A."https://zbmath.org/authors/?q=ai:mba.alphonseSummary: Let \((G,\omega)\) be a presymplectic groupoid. In this paper we characterize the infinitesimal counter part of the tangent presymplectic groupoid of higher order, \((T^rG,\omega^{(c)})\) where \(T^rG\) is the tangent groupoid of higher order and \(\omega^{(c)}\) is the complete lift of higher order of presymplectic form \(\omega\).On a theorem of Calabi for Bakry-Émery Ricci tensor.https://zbmath.org/1449.530382021-01-08T12:24:00+00:00"Li, Guanxun"https://zbmath.org/authors/?q=ai:li.guanxun"Liu, Han"https://zbmath.org/authors/?q=ai:liu.han"Zhang, Shijin"https://zbmath.org/authors/?q=ai:zhang.shijin"Zheng, Yi"https://zbmath.org/authors/?q=ai:zheng.yiSummary: In this paper, the authors obtain two theorems about the estimate for the integral of the Bakry-Émery Ricci curvature along a geodesic on a Riemannian manifold. As an application, the authors obtain sufficient conditions for compactness. They are as generalizations of one theorem of Calabi.Rigidity of complete non-compact gradient expanding Ricci solitons.https://zbmath.org/1449.530412021-01-08T12:24:00+00:00"Chen, Jiarui"https://zbmath.org/authors/?q=ai:chen.jiarui"Liu, Jiancheng"https://zbmath.org/authors/?q=ai:liu.jianchengSummary: By using the existing rigidity theorem of gradient Ricci solitons, we discussed complete non-compact gradient expanding Ricci solitons. Under the conditions that the Ricci curvature was non-negative, the radial curvature vanished and the fourth order divergence of the Weyl tensor was non-negative, we obtained its rigidity result.On a class of Randers metrics of scalar flag curvature.https://zbmath.org/1449.530172021-01-08T12:24:00+00:00"Cheng, Xinyue"https://zbmath.org/authors/?q=ai:cheng.xinyue"Wu, Shasha"https://zbmath.org/authors/?q=ai:wu.shasha"Huang, Qinrong"https://zbmath.org/authors/?q=ai:huang.qinrongSummary: One of the important topics in Finsler geometry is to study and characterize the Finsler metrics of scalar flag curvature, and classifying and characterizing the Randers metrics of scalar flag curvature are still a significant open problem in Finsler geometry. Under the condition that \(\beta\) is a Killing 1-form with respect to \(\alpha\) and a certain condition on \(\Xi\)-curvature, Randers metrics of scalar flag curvature can be characterized. Let \(F\) be a Randers metric of scalar flag curvature on an \(n\)-dimensional manifold \(M (n \ge 3)\). If \(\beta\) is a Killing 1-form with respect to \(\alpha\) and \(F\) satisfies a certain condition on \(\Xi\)-curvature, then \(F\) is of constant flag curvature. The structures of Randers metrics with the conditions mentioned above can be determined completely when dimension \({n \ge 3}\).New aspects of two Hessian-Riemannian metrics in plane.https://zbmath.org/1449.530232021-01-08T12:24:00+00:00"Crâşmăreanu, Mircea"https://zbmath.org/authors/?q=ai:crasmareanu.mirceaSummary: Due to the importance of Hessian structures we express some algebraic and geometric features of two such semi-Riemannian metrics in dimension two. For this purpose we use the separable coordinate systems of the Euclidean plane. Several properties are expressed with the Pauli matrices and their associated quadratic forms.Almost contact B-metric manifolds with curvature tensors of Kähler type.https://zbmath.org/1449.530202021-01-08T12:24:00+00:00"Manev, Mancho"https://zbmath.org/authors/?q=ai:manev.mancho"Ivanova, Miroslava"https://zbmath.org/authors/?q=ai:ivanova.miroslavaSummary: On 5-dimensional almost contact B-metric manifolds, the form of any \(\varphi\)-Kähler-type tensor (i.e. a tensor satisfying the properties of the curvature tensor of the Levi-Civita connection in the special class of the parallel structures on the manifold) is determined. The associated 1-forms are derived by the scalar curvatures of the \(\varphi\)-Kähler-type tensor for the \(\varphi\)-canonical connection on the manifolds from the main classes with closed associated 1-forms.Invariant submanifolds of LP-Sasakian manifolds.https://zbmath.org/1449.530302021-01-08T12:24:00+00:00"Venkatesha, Venkatesha"https://zbmath.org/authors/?q=ai:venkatesha.venkatesha"Basavarajappa, Shanmukha"https://zbmath.org/authors/?q=ai:basavarajappa.shanmukhaSummary: The object of the present paper is to study some geometric conditions for an invariant submanifold of an LP-Sasakian manifold to be totally geodesic. Further we consider concircular curvature tensor satisfying some geometric conditions of an invariant submanifold of an LP-Sasakian manifold to be totally geodesic. In extension, we build an example of LP-Sasakian manifold to verify our main result totally geodesic.Magnetic and slant curves in Kenmotsu manifolds.https://zbmath.org/1449.530272021-01-08T12:24:00+00:00"Pandey, P. K."https://zbmath.org/authors/?q=ai:pandey.prashant-k|pandey.pradeep-kumar|pandey.prabin-kumar|pandey.pradumn-kumar|pandey.pradip-k"Mohammad, S."https://zbmath.org/authors/?q=ai:mohammad.saleh|mohammad.s-a|mohammad.seyed|mohammad.shahid|mohammad.salah-uddin|mohammad.saifSummary: Motivated by the recent studies of the magnetic curves in quasi-Sasakian, Sasakian, and Cosymplectic manifolds, in this article we investigate the magnetic trajectories with respect to contact magnetic fields in Kenmotsu manifolds. Moreover, we study the slant curves, torsion and curvature in Kenmotsu manifolds.The Cauchy problems for dissipative hyperbolic mean curvature flow.https://zbmath.org/1449.580062021-01-08T12:24:00+00:00"Lv, Shixia"https://zbmath.org/authors/?q=ai:lv.shixia"Wang, Zenggui"https://zbmath.org/authors/?q=ai:wang.zengguiSummary: In this paper, we investigate initial value problems for hyperbolic mean curvature flow with a dissipative term. By means of support functions of a convex curve, a hyperbolic Monge-Ampère equation is derived, and this equation could be reduced to the first order quasilinear systems in Riemann invariants. Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems, we discuss lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow.Associated curves of non-lightlike curves due to the Bishop frame of type-1 in Minkowski 3-space.https://zbmath.org/1449.530062021-01-08T12:24:00+00:00"Unluturk, Yasin"https://zbmath.org/authors/?q=ai:unluturk.yasin"Yilmaz, Süha"https://zbmath.org/authors/?q=ai:yilmaz.suha"Çimdiker, Muradiye"https://zbmath.org/authors/?q=ai:cimdiker.muradiye"Şimşek, Sinem"https://zbmath.org/authors/?q=ai:simsek.sinem-nSummary: In this study, we define $M_1,M_2$-direction curves and $M_1,M_2$-donor curves of non-lightlike curve $\gamma$ via the Bishop frame in $E^3_1$. We give some relations about the forementioned curves via the link of the Frenet and Bishop frames. We study the condition for associated curves to be slant helices via the Bishop frame. After defining the spherical indicatrices of associated curves, we obtain some relations between associated curves and their spherical indicatrices in terms of the frames used in the present work.Some recurrent normal Jacobi operators on real hypersurfaces in complex two-plane Grassmannians.https://zbmath.org/1449.530462021-01-08T12:24:00+00:00"Wang, Yaning"https://zbmath.org/authors/?q=ai:wang.yaningSummary: In this paper, we prove that there are no Hopf hypersurfaces in complex two-plane Grassmannians \(G_2(\mathbb{C}^{m+2})\) such that the normal Jacobi operator is generalized \(\mathfrak{F}\)-recurrent, where \(\mathfrak{F} = \text{span}\{\xi,\xi_1,\xi_2,\xi_3\}\). We also prove that there are no Hopf real hypersurfaces in \(G_2(\mathbb{C}^{m+2})\) such that the normal Jacobi operator is \(\mathcal{D}^{\perp}\)-recurrent and the Hopf principal curvature is invariant along the Reeb flow, where \(\mathcal{D}^{\perp}=\text{span}\{\xi_1,\xi_2,\xi_3\}\).Geodesics and geodesic circles in a geodesically convex surface: a sub-mixing property.https://zbmath.org/1449.530372021-01-08T12:24:00+00:00"Innami, Nobuhiro"https://zbmath.org/authors/?q=ai:innami.nobuhiro"Kondo, Toshiki"https://zbmath.org/authors/?q=ai:kondo.toshikiSummary: Let \(M\) be an orientable finitely connected and geodesically convex Finsler surface with genus \(g\ge 1\). We prove that if all geodesics in \(M\) are reversible, then for any number \(\varepsilon>0\) and for any points \(p,q\in M\), there exists a number \(R > 0\) such that any geodesic circle with center \(p\) and radius \(t\) meets the \(\varepsilon\)-ball with center \(q\) for any \(t > R\). Most of the proofs do not use the reversibility assumption for geodesics.Compatibility conditions on the reduced Poisson-Lie group.https://zbmath.org/1449.530342021-01-08T12:24:00+00:00"Aloui, Foued"https://zbmath.org/authors/?q=ai:aloui.foued"Zaalani, Nadhem"https://zbmath.org/authors/?q=ai:zaalani.nadhemSummary: Let \( (G, {\Lambda_G}, {{\langle, \rangle}_G})\) be a Poisson Lie-group equipped with a left invariant Riemannian metric and \(H\) a normal and closed coisotropic subgroup of \(G\). In this paper, we give necessary and sufficient conditions under which the compatibility conditions between the Poisson tensor \({\Lambda_G}\) and the metric \({{\langle, \rangle}_G}\) on \(G\) remain verified on the reduced Poisson-Lie group \( (G/H, {\Lambda_{G/H}}, {{\langle, \rangle}_{G/H}})\). In particular, if the immersed dual group \( (G/H)^*\) is totally geodesic in \({G^*}\), then these conditions are satisfied.Real bisectional curvature, Miyaoka-Yau inequality and Kähler-Ricci flows.https://zbmath.org/1449.530472021-01-08T12:24:00+00:00"Tang, Kai"https://zbmath.org/authors/?q=ai:tang.kaiSummary: In this article, we prove that if \( (X, h)\) is a compact Hermitian manifold with nonpositive real bisectional curvature and \(X\) also admits a Kähler metric, then the Miyaoka-Yau inequality holds. In addition, we can also give an existence time estimate of the Kähler-Ricci flow when there exists a Hermitian metric with real bisectional curvature bounded from above by a positive constant.Multiplicity of generalized Yamabe equations on Riemannian manifolds and applications to Emden-Fowler problems.https://zbmath.org/1449.530402021-01-08T12:24:00+00:00"Liao, Fangfang"https://zbmath.org/authors/?q=ai:liao.fangfang"Heidarkhani, Shapour"https://zbmath.org/authors/?q=ai:heidarkhani.shapour"Afrouzi, Ghasem A."https://zbmath.org/authors/?q=ai:afrouzi.ghasem-alizadeh"Roudbari, Sina Pourali"https://zbmath.org/authors/?q=ai:pourali-roudbari.sinaSummary: In this paper, we study the existence of multiple weak solutions for generalized Yamabe equations on Riemannian manifolds. As applications, we consider the Emden-Fowler equations involving sublinear terms at infinity.Topological loops with six-dimensional solvable multiplication groups having five-dimensional nilradical.https://zbmath.org/1449.220032021-01-08T12:24:00+00:00"Figula, Ágota"https://zbmath.org/authors/?q=ai:figula.agota"Ficzere, Kornélia"https://zbmath.org/authors/?q=ai:ficzere.kornelia"Al-Abayechi, Ameer"https://zbmath.org/authors/?q=ai:al-abayechi.ameerSummary: Using connected transversals we determine the six-dimensional indecomposable solvable Lie groups with five-dimensional nilradical and their subgroups which are the multiplication groups and the inner mapping groups of three-dimensional connected simply connected topological loops. Together with this result we obtain that every six-dimensional indecomposable solvable Lie group which is the multiplication group of a three-dimensional topological loop has one-dimensional centre and two- or three-dimensional commutator subgroup.Classification of time-like conformal homogeneous hypersurfaces with two distinct curvatures in four-dimensional Lorentzian space.https://zbmath.org/1449.530322021-01-08T12:24:00+00:00"Lin, Yanbin"https://zbmath.org/authors/?q=ai:lin.yanbinSummary: A hypersurface \(x (M_1^3)\) in Lorentzian space \(\mathbb{R}_1^4\) is called conformal homogeneous, if for any two points \(p, q\) on \(M_1^3\), there exists \(\sigma\), a conformal transformation of \(\mathbb{R}_1^4\), such that \(\sigma (x (p)) = x (q)\), \(\sigma (x (M_1^3)) = x (M_1^3)\). In this paper, we mainly study the time-like conformal hypersurfaces \(x (M_1^3)\), where the shape operators are diagonal and have two different principal curvatures in \(\mathbb{R}_1^4\). First, by introducing the conformal invariant metric \({g_c}\), the canonical lift \(Y\), conformal tangent frame \({E_i}\) and conformal normal frame \(\xi\), we derive a complete conformal invariant system \(\{{E_1}, {E_2}, {E_3}\}\) for time-like hypersurface in \(\mathbb{R}_1^4\). Then we obtain classification theorems for time-like conformal homogeneous hypersurfaces by providing a series of examples for non-dupin hypersurface, together with the corresponding conformal transformation subgroups.The conformal invariance of the dually flat \(\left({\alpha,\beta} \right)\)-metrics.https://zbmath.org/1449.530162021-01-08T12:24:00+00:00"Cheng, Xinyue"https://zbmath.org/authors/?q=ai:cheng.xinyue"Huang, Qinrong"https://zbmath.org/authors/?q=ai:huang.qinrong"Wu, Shasha"https://zbmath.org/authors/?q=ai:wu.shashaSummary: We study the conformal transformations between two \(\left({\alpha, \beta} \right)\)-metrics. We prove that, if \(F\) is a locally dually flat regular \(\left({\alpha, \beta} \right)\)-metric and is conformally related to \(\bar F\), that is, \(\bar F = {e^{\sigma \left(x \right)}}F\), then \(\bar F\) is also a locally dually flat \(\left({\alpha, \beta} \right)\)-metric if and only if the conformal transformation is a homothety. Further, in the case with singularity, we prove that any conformal transformation between two locally dually flat general Kropina metrics must be a homothety.Duality equations on a 4-manifold of conformal torsion-free connection and some of their solutions for the zero signature.https://zbmath.org/1449.530312021-01-08T12:24:00+00:00"Krivonosov, Leonid Nikolaevich"https://zbmath.org/authors/?q=ai:krivonosov.leonid-nikolaevich"Lukyanov, Vyacheslav Anatol'evich"https://zbmath.org/authors/?q=ai:lukyanov.vyacheslav-anatolevichSummary: On a 4-manifold of conformal torsion-free connection with zero signature \(( --++)\) we found conditions under which the conformal curvature matrix is dual (self-dual or anti-self-dual). These conditions are 5 partial differential equations of the 2nd order on 10 coefficients of the angular metric and 4 partial differential equations of the 1st order, containing also 3 coefficients of external 2-form of charge. (External 2-form of charge is one of the components of the conformal curvature matrix.) Duality equations for a metric of a diagonal type are composed. They form a system of five second-order differential equations on three unknown functions of all four variables. We found several series of solutions for this system. In particular, we obtained all solutions for a logarithmically polynomial diagonal metric, that is, for a metric whose coefficients are exponents of polynomials of four variables.On the fundamental group of complete noncompact Riemannian manifolds.https://zbmath.org/1449.530352021-01-08T12:24:00+00:00"Chen, Aiyun"https://zbmath.org/authors/?q=ai:chen.aiyun"Xue, Qiong"https://zbmath.org/authors/?q=ai:xue.qiong"Xiao, Xiaofeng"https://zbmath.org/authors/?q=ai:xiao.xiaofengSummary: We study the topology of complete noncompact Riemannian manifolds with Ricci curvature satisfies \({\mathrm{Ric}}_M \ge - (n - 1)k (k > 0)\). By using the uniform estimates for the distance from a point to halfway point of minimal geodesics, we prove that a manifold with linear diameter growth has a finitely generated fundamental group.On shrinking gradient 4-D Ricci solitons with sectional curvature bounded above.https://zbmath.org/1449.530512021-01-08T12:24:00+00:00"Zhang, Zhuhong"https://zbmath.org/authors/?q=ai:zhang.zhuhongSummary: The geometric properties of 4-D shrinking gradient solitons are studied with the standard maximum principle, and an important curvature estimate is obtained. More precisely, on a compact 4-D shrinking gradient soliton, if the sectional curvature has a suitable upper bound, the Ricci curvature should be nonnegative. On a noncompact soliton, if the scalar curvature is bounded and has a positive lower bound, the Ricci curvature of the soliton should also be nonnegative.On the translation hypersurfaces with Gauss map \(G\) satisfying \(\Delta G=AG\).https://zbmath.org/1449.530142021-01-08T12:24:00+00:00"Şekerci, G. Aydın"https://zbmath.org/authors/?q=ai:sekerci.gulsah-aydin"Sevinç, S."https://zbmath.org/authors/?q=ai:sevinc.sibel|sevinc.suleyman"Çöken, A. C."https://zbmath.org/authors/?q=ai:coken.abdilkadir-ceylanSummary: It is a known fact that a translation hypersurface is obtained by combination of any three curves in the 4-dimensional Euclidean space. We examine a special situation where the Gauss map of a translation hypersurface satisfies the condition \( \Delta G=AG \) where \( \Delta \) represents the Laplace operator and \( A \) is a \( 4\times 4 \)-real matrix. Our result is that such a translation hypersurface is one of the following three hypersurfaces: the hypersurface of translation surface and a constant vector along this surface, the hyperplane, the hypersurface \( \Sigma\times \mathbb{R} \) where \( \Sigma \) is a translation surface.Non-degenerate hypersurfaces of semi-Riemannian manifold with quarter-symmetric metric connection.https://zbmath.org/1449.530392021-01-08T12:24:00+00:00"Xu, Jingbo"https://zbmath.org/authors/?q=ai:xu.jingbo"Cheng, Xiaoliang"https://zbmath.org/authors/?q=ai:cheng.xiaoliangSummary: Using the equations of Gauss and Weingarten with respect to the Levi-Civita connection, we gave the equations of Gauss and Weingarten for a non-degenerate hypersurface of a semi-Riemannian manifold with a quarter-symmetric metric connection, and obtained the Gauss curvature equation and Codazzi-Mainardi equation for this kind of hypersurface. We could further study the properties of more general connection by using this result.Non-Riemannian Einstein-Randers metrics on \({E_6}/\mathrm{SU} (5)\) and \({E_6}/\mathrm{SU} (2)\).https://zbmath.org/1449.530432021-01-08T12:24:00+00:00"Li, Xiaosheng"https://zbmath.org/authors/?q=ai:li.xiaosheng"Chen, Chao"https://zbmath.org/authors/?q=ai:chen.chao"Hu, Yuwang"https://zbmath.org/authors/?q=ai:hu.yuwangSummary: The homogeneous spaces \({E_6}/SU (5)\) and \({E_6}/SU (2)\) are firstly proved to admit invariant Einstein metrics. Then it's shown that they admit non-Riemannian Einstein-Randers metrics.Curvature in Hilbert geometries.https://zbmath.org/1449.530052021-01-08T12:24:00+00:00"Kurusa, Árpád"https://zbmath.org/authors/?q=ai:kurusa.arpadSummary: We provide more transparent proofs for the facts that the curvature of a Hilbert geometry in the sense of Busemann can not be non-negative and a point of non-positive curvature is a projective center of the Hilbert geometry. Then we prove that the Hilbert geometry has non-positive curvature at its projective centers, and that a Hilbert geometry is a Cayley-Klein model of Bolyai's hyperbolic geometry if and only if it has non-positive curvature at every point of its intersection with a hyperplane. Moreover a 2-dimensional Hilbert geometry is a Cayley-Klein model of Bolyai's hyperbolic geometry if and only if it has two points of non-positive curvature and its boundary is twice differentiable where it is intersected by the line joining those points of non-positive curvature.A new class of metric \(f\)-manifolds.https://zbmath.org/1449.530222021-01-08T12:24:00+00:00"Alegre, Pablo"https://zbmath.org/authors/?q=ai:alegre.pablo"Fernández, Luis M."https://zbmath.org/authors/?q=ai:fernandez.luis-m"Prieto-Martín, Alicia"https://zbmath.org/authors/?q=ai:prieto-martin.aliciaSummary: We introduce a new general class of metric \(f\)-manifolds which we call (almost) trans-\(S\)-manifolds and includes \(S\)-manifolds, \(C\)-manifolds, \(s\)-th Sasakian manifolds and generalized Kenmotsu manifolds studied previously. We prove their main properties and we present many examples which justify their study.Characterization of the cross-linked fibrils under axial motion constraints with graphs.https://zbmath.org/1449.530022021-01-08T12:24:00+00:00"Nagy Kem, Gyula"https://zbmath.org/authors/?q=ai:nagy-kem.gyulaSummary: The filament networks play a significant role in biomaterials as structural stability and transmit mechanical signs. Introducing a 3D mechanical model for the infinitesimal motion of cross-linked fibrils under axial motion constraints, we provide a graph theoretical model and give the characterization of the flexibility and the rigidity of this framework. The connectedness of the graph \(G(v,r)\) of the framework in some cases characterizes the flexibility and rigidity of these structures. In this paper, we focus on the kinematical properties of fibrils and proof the next theorem for generic nets of fibrils that are cross-linked by another type of fibrils. ``If the fibrils and the bars are generic positions, the structure will be rigid if and only if each of the components of \(G(v,r)\) has at least one circuit.'' We offer some conclusions, including perspectives and future developments in the frameworks of biostructures as microtubules, collagens, celluloses, actins, other polymer networks, and composite which inspired this work.A global rigidity theorem for the length of concircular curvature tensor on the locally conformally symmetric Riemannian manifold.https://zbmath.org/1449.530362021-01-08T12:24:00+00:00"Hu, Shipei"https://zbmath.org/authors/?q=ai:hu.shipeiSummary: In this paper, we study the locally conformally symmetric closed Riemannian manifold, and establish a global rigidity theorem for the length of the concircular curvature vector.Radius and concurrent vector fields in spray and Finsler geometry.https://zbmath.org/1449.530032021-01-08T12:24:00+00:00"Crampin, Mike"https://zbmath.org/authors/?q=ai:crampin.mikeThis paper concerns with types of vector fields, namely concurrent and of radius type, in two settings: Finsler geometry and more generally spray geometry. A radius vector field is one which is both concurrent and affine. An important particular case of spray geometry is that provided by a fixed (and symmetric) linear connection. Hence, a major theme in this study is the role of radius vector fields in the theory of projective connections. A main result is a rigidity one: a connected spray manifold endowed with a complete radius vector field possesing a zero is diffeomorphic to the Euclidean space \(\mathbb{R}^n\) with its standard flat spray and canonical Euler radius vector field \(\Lambda =x^a\frac{\partial}{\partial x^a}\).
Reviewer: Mircea Crâşmăreanu (Iaşi)Ricci solitons on three dimensional generalized Sasakian space forms with quasi Sasakian metric.https://zbmath.org/1449.530292021-01-08T12:24:00+00:00"Sarkar, Avijit"https://zbmath.org/authors/?q=ai:sarkar.avijit"Biswas, Gour Gopal"https://zbmath.org/authors/?q=ai:biswas.gour-gopalSummary: The object of the present paper is to study Ricci solitons on three dimensional generalized Sasakian space forms with quasi Sasakian metric. We have also studied gradient Ricci solitons on such manifolds. Examples have been given.Associated curves of the spacelike curve via the Bishop frame of type-2 in \(\mathbb{E}^3_1\).https://zbmath.org/1449.530012021-01-08T12:24:00+00:00"Ünlütürk, Yas\{in"https://zbmath.org/authors/?q=ai:unluturk.yasin"Yilmaz, Süha"https://zbmath.org/authors/?q=ai:yilmaz.suhaSummary: The objective of the study in this paper is to define \(M_1,M_2\)-direction curves and \(M_1,M_2\)-donor curves of the spacelike curve \(\gamma\) via the Bishop frame of type-2 in \(\mathbb{E}^3_1\). We obtain necessary and sufficient conditions when the associated curves to be slant helices and general helices via the Bishop frame of type-2 in \(\mathbb{E}^3_1\). After defining the spherical indicatrices of the associated curves, we obtain some relations between associated curves and their spherical indicatrices in terms of the frames used in the present work.Spherically symmetric Finsler metrics with scalar flag curvature.https://zbmath.org/1449.530182021-01-08T12:24:00+00:00"Liu, Min"https://zbmath.org/authors/?q=ai:liu.min"Song, Weidong"https://zbmath.org/authors/?q=ai:song.weidongSummary: Finsler geometry is Riemannian geometry without quadratic restriction, and the projectively flat Finsler metrics are very important in Finsler geometry. Here a new example of projectively flat spherically symmetric Finsler metric was given by investigating a PDE equivalent, and its flag curvature was obtained.The almost Einstein operator for \((2,3,5)\) distributions.https://zbmath.org/1449.530332021-01-08T12:24:00+00:00"Sagerschnig, Katja"https://zbmath.org/authors/?q=ai:sagerschnig.katja"Willse, Travis"https://zbmath.org/authors/?q=ai:willse.travisA \((2,3,5)\) distribution on a 5-manifold is a \((2,3)\)-plane distribution which can be viewed as maximally non-integrable. This distribution induces a conformal structure of signature \((2,3)\). In the paper the corresponding tractor calculus is developed. The paper is motivated by the question whether the conformal structure contains an Einstein metric.
Reviewer: Hans-Bert Rademacher (Leipzig)Three dimensional near-horizon metrics that are Einstein-Weyl.https://zbmath.org/1449.530112021-01-08T12:24:00+00:00"Randall, Matthew"https://zbmath.org/authors/?q=ai:randall.matthewThe paper investigates three-dimensional near-horizon metrics \(g=2dydr+2rh(x)dxdy+r^2f(x)dy^2+dx^2\) whose associate conformal structure is Einstein-Weyl. This means that there exists a symmetric connection \(D\) preserving the structure \([g]\) and such that \(\operatorname{Ric}_D^{sym}=\Lambda g\) for some function \(\Lambda\). By imposing a certain ansatz on the covector field \(g^{-1}Dg\) the author finds several classes of solutions, among which two are particular remarkable: one that is defined by some solutions of the dispersionless Kadomtsev-Petviashvili equation and one that is defined by some solutions of the dispersionless Hirota type, also known as hyper-CR. The work is well written and is of interest for both communities studying near horizon geometries and studying Einstein-Weyl geometry and dispersionless integrable systems.
Reviewer: Boris S. Kruglikov (Tromsø)Almost c-spinorial geometry.https://zbmath.org/1449.530072021-01-08T12:24:00+00:00"Púček, Roland"https://zbmath.org/authors/?q=ai:pucek.rolandAlmost c-spinorial geometry arises as an example of the metrisability problem for parabolic geometries. It is a complex analogue of real spinorial geometry. The author defines such type of parabolic geometry, and he discusses its underlying geometry and its homogeneous model. He computes the irreducible components of the harmonic curvature and discusses the conditions for regularity. He also describes the linearisation of the metrisability problem for Hermitian and skew-Hermitian metrics, states the corresponding first BGG equations and presents explicit formulae for their solutions on the homogeneous model.
Reviewer: Andreas Arvanitoyeorgos (Patras)Curvature properties of two Naveira classes of Riemannian product manifolds.https://zbmath.org/1449.530242021-01-08T12:24:00+00:00"Gribacheva, Dobrinka"https://zbmath.org/authors/?q=ai:gribacheva.dobrinka-kostadinovaSummary: The main aim of the present work is to obtain some curvature properties of the manifolds from two classes of Riemannian product manifolds. These classes are two basic classes from Naveira classification of Riemannian almost product manifolds.Invariant Einstein metrics on some generalized flag manifolds with six isotropy summands.https://zbmath.org/1449.530442021-01-08T12:24:00+00:00"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.3"Qin, Huajun"https://zbmath.org/authors/?q=ai:qin.huajun"Zhao, Guosong"https://zbmath.org/authors/?q=ai:zhao.guosongSummary: There are two difficulties to obtain invariant Einstein metrics on generalized flag manifolds \(G/K\), one is how to compute non-zero structure constants of the flag manifolds, the other is how to compute Gröbner bases of the system of Einstein equations. In this paper, the authors compute non-zero structure constants and get Gröbner bases of the system of Einstein equations by using the software Maple. In this way the authors obtain invariant Einstein metrics on the flag manifolds \({F_4}/{U^2} (1) \times SU (3)\), \({E_6}/{U^2} (1) \times SU (3) \times SU (3)\), \({E_7}/{U^2} (1) \times SU (2) \times SU (5)\), \({E_7}/{U^2} (1) \times SU (6)\), \({E_7}/{U^2} (1) \times SU (2) \times SO (8)\) and \({E_8}/{U^2} (1) \times {E_6}\) respectively.Commutativity of torsion and normal Jacobi operators on real hypersurfaces in the complex quadric.https://zbmath.org/1449.530282021-01-08T12:24:00+00:00"Pérez, Juan de Dios"https://zbmath.org/authors/?q=ai:perez-jimenez.juan-de-diosAs the author mentions in the abstract of his paper, ``on a real hypersurface in the complex quadric we can consider the Levi-Civita connection and, for any non-zero real constant \(k\), the \(k\)-th generalized Tanaka-Webster connection. Associated to this connection we can define a differential operator whose difference with Lie derivative is the torsion operator of the \(k\)-th generalized Tanaka-Webster connection''. In the paper under review, the author proves ``the non-existence of real hypersurfaces in the complex quadric for which the torsion operators commute with the normal Jacobi operator of the real hypersurface''.
Reviewer: Radu Iordănescu (Bucureşti)\(\phi \)-pseudo sphere Gauss maps of Lorentzian hypersurfaces in anti de Sitter space.https://zbmath.org/1449.530152021-01-08T12:24:00+00:00"Miao, Jiajing"https://zbmath.org/authors/?q=ai:miao.jiajing"Liu, Haiming"https://zbmath.org/authors/?q=ai:liu.haimingSummary: In this paper, we prove the existence of \(\varphi \)-pseudo sphere Gauss maps of Lorentzian hypersurfaces in anti de Sitter space by Legendrian dual theorem, and we preliminarily establish the slant geometry of Lorentzian hypersurfaces in anti de Sitter space. The basic theorem of slat geometry is proved and the classification of \({\phi^\pm}\)-totally umbilical hypersurface and \({\phi^\pm}\)-anti de Sitter Weingarten formula are given.