Recent zbMATH articles in MSC 53C25https://zbmath.org/atom/cc/53C252021-06-15T18:09:00+00:00WerkzeugPeriodic solutions to Navier-Stokes equations on non-compact Einstein manifolds with negative curvature.https://zbmath.org/1460.352632021-06-15T18:09:00+00:00"Nguyen, Thieu Huy"https://zbmath.org/authors/?q=ai:nguyen-thieu-huy."Pham, Truong Xuan"https://zbmath.org/authors/?q=ai:pham.truong-xuan"Vu, Thi Ngoc Ha"https://zbmath.org/authors/?q=ai:vu-thi-ngoc-ha."Vu, Thi Mai"https://zbmath.org/authors/?q=ai:vu.thi-maiSummary: Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold \((\mathbf{M},g)\) with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on \((\mathbf{M},g)\). Our method is based on the dispersive and smoothing properties of the semigroup generated by the linearized Stokes equations to construct a bounded (in time) solution of the nonhomogeneous Stokes equation and on the ergodic method to obtain the periodic solution to Stokes equation. Then, using the fixed point arguments, we can pass from the Stokes equations to Navier-Stokes equations to obtain periodic solutions to NSE on the Einstein manifold \((\mathbf{M},g)\). We also prove the stability of the periodic solution.A new class of golden Riemannian manifold.https://zbmath.org/1460.530302021-06-15T18:09:00+00:00"Beldjilali, Gherici"https://zbmath.org/authors/?q=ai:beldjilali.ghericiSummary: In this paper, we introduce a new class of almost Golden Riemannian structures and study their essential examples as well as their fundamental properties. Next, we investigate a particular type belonging to this class and we establish some basic results for Riemannian curvature tensor and the sectional curvature. Concrete examples are given.Regularity of fully non-linear elliptic equations on Kähler cones.https://zbmath.org/1460.351482021-06-15T18:09:00+00:00"Yuan, Rirong"https://zbmath.org/authors/?q=ai:yuan.rirongSummary: We derive quantitative boundary estimates, and then solve the Dirichlet problem for a general class of fully non-linear elliptic equations on annuli of Kähler cones over closed Sasakian manifolds. This extends extensively a result concerning the geodesic equations in the space of Sasakian metrics due to \textit{P. Guan} and \textit{X. Zhang} [Adv. Math. 230, No. 1, 321--371 (2012; Zbl 1245.58016)]. Our results show that the solvability is deeply affected by the transverse Kähler structures of Sasakian manifolds. We also discuss possible extensions of the results to equations with right-hand side depending on unknown solutions.Nonexistence for hyperbolic problems on Riemannian manifolds.https://zbmath.org/1460.353642021-06-15T18:09:00+00:00"Monticelli, Dario D."https://zbmath.org/authors/?q=ai:monticelli.dario-daniele"Punzo, Fabio"https://zbmath.org/authors/?q=ai:punzo.fabio"Squassina, Marco"https://zbmath.org/authors/?q=ai:squassina.marcoSummary: We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.Conformal vector fields and Ricci soliton structures on natural Riemann extensions.https://zbmath.org/1460.530142021-06-15T18:09:00+00:00"Abbassi, Mohamed Tahar Kadaoui"https://zbmath.org/authors/?q=ai:abbassi.mohamed-tahar-kadaoui"Amri, Noura"https://zbmath.org/authors/?q=ai:amri.noura"Bejan, Cornelia-Livia"https://zbmath.org/authors/?q=ai:bejan.cornelia-liviaSummary: The framework of the paper is the phase universe, described by the total space of the cotangent bundle of a manifold \(M\), which is of interest for both mathematics and theoretical physics. When \(M\) carries a symmetric linear connection, then \(T^*M\) is endowed with a semi-Riemannian metric, namely the classical Riemann extension, introduced by Patterson and Walker and then by Willmore. We consider here a generalization provided by Sekizawa and Kowalski of this metric, called the natural Riemann extension, which is also a metric of signature \((n, n)\). We give the complete classification of conformal and Killing vector fields with respect to an arbitrary natural Riemann extension. Ricci soliton is a topic that has been increasingly studied lately. Necessary and sufficient conditions for the phase space to become a Ricci soliton (or Einstein) are given at the end.Wasserstein Riemannian geometry on statistical manifold.https://zbmath.org/1460.530422021-06-15T18:09:00+00:00"Ogouyandjou, Carlos"https://zbmath.org/authors/?q=ai:ogouyandjou.carlos"Wadagni, Nestor"https://zbmath.org/authors/?q=ai:wadagni.nestorSummary: In this paper, we study some geometric properties of statistical manifold equipped with the Riemannian Otto metric which is related to the \(\mathcal L ^2\) -Wasserstein distance of optimal mass transport. We construct some \(\alpha \) -connections on such manifold and we prove that the proposed connections are torsion-free and coincide with the Levi-Civita connection when \(\alpha = 0\). In addition, the exponentialy families and the mixture families are shown to be respectively (1)-flat and \((-1)\)-flat.Riemannian manifolds and homogeneous geodesics.https://zbmath.org/1460.530012021-06-15T18:09:00+00:00"Berestovskii, Valerii"https://zbmath.org/authors/?q=ai:berestovskii.valerii-nikolaevich"Nikonorov, Yurii"https://zbmath.org/authors/?q=ai:nikonorov.yurii-gA geodesic \(\gamma\) in a Riemannian manifold \((M, g)\) is called homogeneous if it is an orbit of some 1-parameter subgroup \(\gamma (t)\) of the isometry group \(\mathrm{Isom}(M,g)\) of \(M\) (and consequently, \(\gamma\) is an integral curve of a Killing vector field, generated by this subgroup).
The main topic of the present book is the study of homogeneous geodesics in Riemannian manifolds. As it is well known, the study of geodesics in a Riemannian manifold is not an easy matter, hence one looks for simpler, still interesting, expressions of geodesics. Of special importance are Riemannian manifolds \((M, g)\) with the property that every geodesic is homogeneous. These spaces are called geodesic orbit manifolds (GO-manifolds). A Riemannian homogeneous space \((G/H, g)\), where \(g\) is a \(G\)-invariant metric and \(H\) a compact Lie subgroup of a Lie group \(G\) is called geodesic orbit space (GO-space), if any geodesic of \(G/H\) is an orbit of some 1-parameter subgroup of \(G\). The metric \(g\) is then called a GO-metric. The study of GO-spaces was initiated by \textit{O. Kowalski} and \textit{L. Vanhecke} [Boll. Unione Mat. Ital., VII. Ser., B 5, No. 1, 189--246 (1991; Zbl 0731.53046)], and since then it has been expanded by several authors, producing many interesting results as well as serious links to other concepts of differential geometry and Lie theory.
The book can be divided into two parts. The first part includes Chapters 1 to 4. This contains basic material about Riemannian manifolds, Lie groups, isometric flows and Riemannian homogeneous spaces. Possibly not for a first reading on Riemannian geometry, still the material is presented in a very didactic and informative manner. The interested reader can find standard and non-standard results on these topics and alternative or simpler proofs in some cases. For example, a simple proof of S. Myers' theorem on compactness of a complete Riemannian manifold with Ricci curvature bounded below, O'Neill's formulas for Riemannian submersions, some theorems of É.B. Vinberg and B. Kostant about invariant norms, some new formulas for the curvature tensor and Killing fields in a Riemannian manifold, Killing vector fields of constant length, a discussion about the structure of the set of invariant metrics in a homogeneous space, results about compact homogeneous Riemannian manifolds of positive Euler characteristic or positive Ricci curvature, to name a few.
The second part includes Chapters 5 to 7 and these constitute the main core of the book. Chapter 5 includes a review of various results on manifolds with homogeneous geodesics including contributions of the authors. Some of these include, Killing vector fields of constant length and GO-spaces, GO-metrics on spheres, GO-spaces of positive Euler characteristic, and various classification results (e.g., compact GO-spaces with two isotropy summands, GO-metrics on generalized flag manifolds, Ledger-Obata spaces). Chapter 6 is devoted to the study of \(\delta\)-homogeneous Riemannian manifolds (or generalized normal homogeneous). These spaces constitute a proper subclass of GO-spaces with non-negative sectional curvature. Finally, Chapter 7 is devoted to the study of Clifford-Wolf homogeneous Riemannian manifolds and their natural generalizations. Also, the authors study Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional spheres. The two last chapters are of more advanced level, however detailed proofs are provided in most cases. Even proofs of original papers of the authors are given in a more detailed manner.
In summary, the book by V. Berestovskii and Y. Nikonorov is a welcome contribution in homogeneous geometry, which can be appreciated by a wide range of audience, from graduate students to focused researchers.
Reviewer: Andreas Arvanitoyeorgos (Patras)On slant curves in Sasakian Lorentzian 3-manifolds.https://zbmath.org/1460.530212021-06-15T18:09:00+00:00"Lee, Ji-Eun"https://zbmath.org/authors/?q=ai:lee.jieunSummary: In this paper, we study \(C\)-parallel mean curvature vector field and \(C\)-proper mean curvature vector field along a slant Frenet curve in a Sasakian Lorentzian 3-manifold. In particular, we prove that a slant Frenet curve \(\gamma\) in a Sasakian Lorentzian \(3\)-manifold \(M\) satisfying \(\Delta_{\dot{\gamma}} H =0\) is a geodesic or pseudo-helix with \(\kappa^2=\tau^2\). For example, we find slant pseudo-helix in Lorentzian Heisenberg 3-space.Some Ricci solitons on Kenmotsu manifold.https://zbmath.org/1460.530482021-06-15T18:09:00+00:00"Shanmukha, B."https://zbmath.org/authors/?q=ai:shanmukha.b"Venkatesha, V."https://zbmath.org/authors/?q=ai:venkatesha.vishnuvardhana-s-v|venkatesha.venkateshaSummary: In the present frame work, we study the properties of \(\eta \)-Ricci soliton on Kenmotsu manifold and also analysed the generalized gradient Ricci soliton equation satisfying some conditions.Escobar-Yamabe compactifications for Poincaré-Einstein manifolds and rigidity theorems.https://zbmath.org/1460.530442021-06-15T18:09:00+00:00"Chen, Xuezhang"https://zbmath.org/authors/?q=ai:chen.xuezhang"Lai, Mijia"https://zbmath.org/authors/?q=ai:lai.mijia"Wang, Fang"https://zbmath.org/authors/?q=ai:wang.fangSummary: Let \((X^n, g_+)\)\((n \geq 3)\) be a Poincaré-Einstein manifold which is \(C^{3, \alpha}\) conformally compact with conformal infinity \((\partial X, [\hat{g}])\). On the conformal compactification \((\overline{X}, \overline{g} = \rho^2 g_+)\) via a boundary defining function \({\rho}\), there are two types of Yamabe constants: \(Y(\overline{X}, \partial X, [\overline{g}])\) and \(Q(\overline{X}, \partial X, [\overline{g}])\). (See definitions (1) and (2).) In [Geom. Funct. Anal. 27, No. 4, 863--879 (2017; Zbl 1379.53058)], \textit{M. J. Gursky} and \textit{Q. Han} obtained an inequality between \(Y(\overline{X}, \partial X, [\overline{g}])\) and \(Y(\partial X, [\hat{g}])\). In this paper, we first show that the equality holds in Gursky-Han's inequality if and only if \((X^n, g_+)\) is isometric to the standard hyperbolic space \((\mathbb{H}^n, g_{\mathbb{H}})\). Secondly, we derive a conformal invariant inequality between \(Q(\overline{X}, \partial X, [\overline{g}])\) and \(Y(\partial X, [\hat{g}])\), and show the equality holds if and only if \((X^n, g_+)\) is isometric to \((\mathbb{H}^n, g_{\mathbb{H}})\). Based on this, we give a simple proof of the rigidity theorem for Poincaré-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.Lorentzian para-Sasakian manifolds admitting a new type of quarter-symmetric non-metric \(\xi \)-connection.https://zbmath.org/1460.530152021-06-15T18:09:00+00:00"Chaubey, S. K."https://zbmath.org/authors/?q=ai:chaubey.sudhakar-kumar"De, Uday Chand"https://zbmath.org/authors/?q=ai:de.uday-chandSummary: We define a new type of quarter-symmetric non-metric \(\xi \)-connection on an \(LP\)-Sasakian manifold and prove its existence. We provide its application in the general theory of relativity. To validate the existence of the quarter-symmetric non-metric \(\xi \)-connection on an \(LP\)-Sasakian manifold, we give a non-trivial example in dimension \(4\) and verify our results.On quasi-Sasakian \(3\)-manifolds with respect to the Schouten-Van Kampen connection.https://zbmath.org/1460.530472021-06-15T18:09:00+00:00"Perktaş, Selcen Yüksel"https://zbmath.org/authors/?q=ai:perktas.selcen-yuksel"Yildiz, Ahmet"https://zbmath.org/authors/?q=ai:yildiz.ahmetSummary: In this paper we study some soliton types on a quasi-Sasakian 3-manifold with respect to the Schouten-van Kampen connection.Almost contact \(B\)-metric structure on 5-dimensional nilpotent Lie algebras.https://zbmath.org/1460.530312021-06-15T18:09:00+00:00"Bulut, Şenay"https://zbmath.org/authors/?q=ai:bulut.senay"Ermiş, Sevgi Enveş"https://zbmath.org/authors/?q=ai:ermis.sevgi-envesSummary: The classification of almost contact \(B\)-metric manifolds is considered. It is shown that \(\mathcal{D}\)-homothetic deformation of these manifolds in the class \(\mathcal{F}_i\) (\( i=0,1,4,5\)) remains in the same the class \(\mathcal{F}_i \). We study almost contact \(B\)-metric structure on 5-dimensional nilpotent Lie algebras. The class of the left invariant almost contact \(B\)-metric structures on corresponding Lie groups is investigated. Finally, we determine the class of 5-dimensional nilpotent Lie algebras with almost contact \(B\)-metric structure.Indefinite Einstein metrics on nice Lie groups.https://zbmath.org/1460.530452021-06-15T18:09:00+00:00"Conti, Diego"https://zbmath.org/authors/?q=ai:conti.diego"Rossi, Federico A."https://zbmath.org/authors/?q=ai:rossi.federico-albertoSummary: We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension \(\geq 8\).An application of the Duistermaat-Heckman theorem and its extensions in Sasaki geometry.https://zbmath.org/1460.530432021-06-15T18:09:00+00:00"Boyer, Charles P."https://zbmath.org/authors/?q=ai:boyer.charles-p"Huang, Hongnian"https://zbmath.org/authors/?q=ai:huang.hongnian"Legendre, Eveline"https://zbmath.org/authors/?q=ai:legendre.evelineLet \((N,D)\) be a compact contact manifold of Sasaki type of dimension \(2n+1\), with a cooriented rank-\(2n\) contact distribution \(D\). If
\(T\subset \boldsymbol{CR}\) is the maximal torus of automorphisms, the (reduced) Sasaki cone (or Reeb cone) denoted by \(\mathfrak{t}^+\), is an open polyhedral cone in \(\mathfrak{t}=\operatorname{Lie}(T)\) and contains all the \(T\)-invariant Reeb vector fields on \((N,D,J)\) where
\(\boldsymbol{CR}\)-structure \(J\in\operatorname{End}(D)\) (i.e. \(J^2=-\operatorname{id}_D +\) integrability condition).
Using the Duistermaat-Heckman localization formula and an extension of it the authors give rational and explicit expressions of the volume,
the total transversal scalar curvature and the Einstein-Hilbert functionals on the Sasaki cone \(\mathfrak{t}^+\). Furthermore, assuming
\(\xi\in \mathfrak{t}^+\) tends to the boundary of \(\mathfrak{t}^+\), the authors prove the total transversal scalar curvature and the Einstein-Hilbert functional \(H(\xi)\) tend to \(+\infty\). Some properties of \(H(\xi)\) are also studied. Let us mention especially that \(H(\xi)\) attains its minimal value along a ray in the interior of the cone \(\mathfrak{t}^+\), and each Sasaki cone possesses at least one Reeb vector field with vanishing transverse Futaki invariant.
The authors also discuss the effect of positivity (or negativity) of the total transversal scalar curvature onto the transverse geometry.
Reviewer: Neda Bokan (Beograd)Schouten and Vrănceanu connections on golden manifolds.https://zbmath.org/1460.530252021-06-15T18:09:00+00:00"Gök, Mustafa"https://zbmath.org/authors/?q=ai:gok.mustafa"Keleş, Sadık"https://zbmath.org/authors/?q=ai:keles.sadik"Kiliç, Erol"https://zbmath.org/authors/?q=ai:kilic.erolSummary: In this paper, we study two special linear connections, which are called Schouten and Vrănceanu connections, defined by an arbitrary fixed linear connection on a differentiable manifold admitting a golden structure. The golden structure defines two naturally complementary projection operators splitting the tangent bundle into two complementary parts, so there are two globally complementary distributions of the tangent bundle. We examine the conditions of parallelism for both of the distributions with respect to the fixed linear connection under the assumption that it is either the Levi-Civita connection or is not. We investigate the concepts of half parallelism and anti half parallelism for each of the distributions with respect to the Schouten and Vrănceanu connections. We research integrability conditions of the golden structure and its associated distributions from the viewpoint of the Schouten and Vrănceanu connections. Finally, we analyze the notion of geodesicity on golden manifolds in terms of the Schouten and Vrănceanu connections.Gradient Yamabe solitons on multiply warped product manifolds.https://zbmath.org/1460.530462021-06-15T18:09:00+00:00"Karaca, Fatma"https://zbmath.org/authors/?q=ai:karaca.fatmaSummary: We consider gradient Yamabe solitons on multiply warped product manifolds. We find the necessary and sufficient conditions for multiply warped product manifolds to be gradient Yamabe solitons.Quasi-classical approximation for magnetic monopoles.https://zbmath.org/1460.780072021-06-15T18:09:00+00:00"Kordyukov, Yuri A."https://zbmath.org/authors/?q=ai:kordyukov.yuri-a"Taimanov, Iskander A."https://zbmath.org/authors/?q=ai:taimanov.iskander-aA note on \(f\)-biharmonic Legendre curves in \(\mathcal{S}\)-space forms.https://zbmath.org/1460.530562021-06-15T18:09:00+00:00"Güvenç, Şaban"https://zbmath.org/authors/?q=ai:guvenc.sabanSummary: In this paper, we study \(f\)-biharmonic Legendre curves in \(\mathcal{S}\)-space forms. Our aim is to find curvature conditions for these curves and determine their types, i.e., a geodesic, a circle, a helix or a Frenet curve of osculating order \(r\) with specific curvature equations. We also give a proper example of \(f\)-biharmonic Legendre curves in the S-space form \(R^{2m+s}(-3 s)\), with \(m = 2\) and \(s = 2\).