Recent zbMATH articles in MSC 53Dhttps://zbmath.org/atom/cc/53D2022-07-25T18:03:43.254055ZWerkzeugPrincipal actions of stacky Lie groupoidshttps://zbmath.org/1487.140042022-07-25T18:03:43.254055Z"Bursztyn, Henrique"https://zbmath.org/authors/?q=ai:bursztyn.henrique"Noseda, Francesco"https://zbmath.org/authors/?q=ai:noseda.francesco"Zhu, Chenchang"https://zbmath.org/authors/?q=ai:zhu.chenchangThis paper is concerned with, as the title implies, actions of stacky Lie groupoids on differentiable stacks. Ultimately, the authors show that there is a characterization of those actions that are principal (Definition 4.21), i.e., actions whose quotient yields a differentiable stack and whose projection map can be seen as a principal bundle, in terms of the weak representability (Definition 2.2) of the so-called action-projection map (Theorem 5.2). For a bit of context, this is analog to the famous result that a quotient of a Lie group action on a manifold yields a manifold if and only if the action is free and proper. Consequence of this theorem, it follows that the category of differentiable stacks is stable under quotients by Lie groupoids (Corollary 5.3). The authors then move on to apply their theory to prove that the naturally defined Morita equivalence of Stacky Lie groupoids yields an equivalence relation that (faithfully) extends that of Lie groupoids (Theorem 6.10). Concluding, they illustrate this last result by considering transitive Lie algebroids given as rank one extensions of the tangent bundle corresponding to closed 2-forms whose group of periods is not discrete (Section 6.2).
Reviewer: Camilo Andres Angulo Santacruz (Rio de Janeiro)Chen-Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surfacehttps://zbmath.org/1487.140792022-07-25T18:03:43.254055Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Das, Pradeep"https://zbmath.org/authors/?q=ai:das.pradeep"Singh, Anoop"https://zbmath.org/authors/?q=ai:singh.anoopSummary: Let \((X,\,D)\) be an \(m\)-pointed compact Riemann surface of genus at least 2. For each \(x \,\in \, D\), fix full flag and concentrated weight system \(\alpha\). Let \(P \mathcal{M}_{\xi}\) denote the moduli space of semi-stable parabolic vector bundles of rank \(r\) and determinant \(\xi\) over \(X\) with weight system \(\alpha\), where \(r\) is a prime number and \(\xi\) is a holomorphic line bundle over \(X\) of degree \(d\) which is not a multiple of \(r\). We compute the Chen-Ruan cohomology of the orbifold for the action on \(P \mathcal{M}_{\xi}\) of the group of \(r\)-torsion points in \(\text{Pic}^0 (X)\).Effective computations of the Atiyah-Bott formulahttps://zbmath.org/1487.141152022-07-25T18:03:43.254055Z"Muratore, Giosuè"https://zbmath.org/authors/?q=ai:muratore.giosue-emanuele"Schneider, Csaba"https://zbmath.org/authors/?q=ai:schneider.csabaIn this paper, utilizing the Atiyah-Bott residue formula for \(\overline{M}_{0,n}(\mathbb{P}^{n},d)\), the authors compute a large number of Gromov-Witten invariants. They give an extensive table that contains the number of rational contact curves in \(\mathbb{P}^{3}\) up to degree 8 meeting arbitrary linear spaces. They also provide a code which is publicly available.
Reviewer: Vehbi Emrah Paksoy (Fort Lauderdale)On the quantum \(K\)-theory of the quintichttps://zbmath.org/1487.141202022-07-25T18:03:43.254055Z"Garoufalidis, Stavros"https://zbmath.org/authors/?q=ai:garoufalidis.stavros"Scheidegger, Emanuel"https://zbmath.org/authors/?q=ai:scheidegger.emanuelThe authors present a conjecture for the small \(J\) -function and its small linear \(q-\)difference equation expressed linearly in terms of Gopakumar-Vafa invariants. More explicitly, they conjecture that the small \(J\) function of the quintic 3-fold is expressed linearly in terms of GV-invariants by
\[
\frac{1}{1-q}J(Q,q,0)=1+x^{2}\sum_{d,r\geq 1}a(d,r,q^{r})\mbox{GV}_{d}\mbox{Q}^{dr}+x^{3}\sum_{d,r\geq 1}b(d,r,q^{r})\mbox{GV}_{d}\mbox{Q}^{dr}.
\]
The authors also study the consequence of the conjecture and its relations to a conjecture by \textit{H. Jockers} and \textit{P. Mayr} [J. High Energy Phys. 2019, No. 11, Paper No. 11, 21 p. (2019; Zbl 1429.81090)].
Reviewer: Vehbi Emrah Paksoy (Fort Lauderdale)Structure and cohomology of 3-Lie-Rinehart superalgebrashttps://zbmath.org/1487.170082022-07-25T18:03:43.254055Z"Ben Hassine, Abdelkader"https://zbmath.org/authors/?q=ai:ben-hassine.abdelkader"Chtioui, Taoufik"https://zbmath.org/authors/?q=ai:chtioui.taoufik"Mabrouk, Sami"https://zbmath.org/authors/?q=ai:mabrouk.sami"Silvestrov, Sergei"https://zbmath.org/authors/?q=ai:silvestrov.sergei-dIn this paper, the authors introduce the concept of 3-Lie-Rinehart superalgebra over an associative supercommutative superalgebra \(A\) over a field \(\mathbb{K}\) as a tuple \((L,A,[\cdot,\cdot,\cdot],\rho)\), where \(L\) is an \(A\)-module endowed with structure of \(3\)-Lie superalgebra with the even super skew-symmetric trilinear map \([\cdot,\cdot,\cdot]:L\times L\times L\rightarrow L\), so that: (a) \(\rho:L\times L\rightarrow \mathrm{Der}(A)\) is a representation of \((L,[\cdot,\cdot,\cdot])\) on \(A\); (b) \(\rho(ax,y)=(-1)^{\overline{a}\,\overline{x}}\rho(x,ay)=a\rho(x,y)\), for all \(x,y\in\mathcal{H}(L)\) and \(a\in\mathcal{H}(A)\); and (c) the following compatibility condition holds: \([x,y,az]=(-1)^{\overline{a}(\overline{x}+\overline{y})}a[x,y,z]+\rho(x,y)az\), for all \(x,y,z\in \mathcal{H}(L)\) and \(a\in \mathcal{H}(A)\). Here, \(\mathcal{H}(V)\) denotes the set of homogeneous elements of a graded vector space.
Once this concept is introduced, the authors describe some constructions of 3-Lie-Rinehart superalgebras starting with either a Lie-Rinehart superalgebra or an other 3-Lie-Rinehart superalgebra. They also detail the construction of a Lie-Rinehart superalgebras starting from a 3-Lie-Rinehart superalgebra. Furthermore, it is introduced the concepts of module, \(1\)- and \(2\)-cocycle for a 3-Lie-Rinehart superalgebra, which make possible to describe systematically a cohomology complex for this type of superalgebras with coefficients in a module. This cohomology enables the authors to deal with the notion of deformation of a 3-Lie-Rinehart superalgebra. An equivalence relation among these deformations is also studied.
Reviewer: Raúl M. Falcón (Sevilla)Characters, coadjoint orbits and Duistermaat-Heckman integralshttps://zbmath.org/1487.220142022-07-25T18:03:43.254055Z"Alekseev, Anton"https://zbmath.org/authors/?q=ai:alekseev.anton-yu"Shatashvili, Samson L."https://zbmath.org/authors/?q=ai:shatashvili.samson-lLet \(G\) be a compact Lie group with Lie algebra \(\mathfrak{g}\) and \(\chi_\lambda\) the character of an irreducible representation of \(G\) of highest weight \(\lambda\). Kirillov's character formula expresses \(\chi_\lambda(\exp(h))\) for small \(h\in\mathfrak{g}\) in terms of the Fourier transform of the Liouville measure on the associated coadjoint orbit \(\mathcal{O}_\lambda\subseteq\mathfrak{g}^*\). This formula can also be obtained by applying the Duistermaat-Heckman formula to the oscillatory integral over the symplectic manifold \(\mathcal{O}_\lambda\) defining the Fourier transform, which is why the authors refer to these integrals as \emph{Duistermaat-Heckman integrals}. As a consequence, the asymptotic behaviour of the rescaled character \(\chi_{k\lambda}(\exp(h/k))\) as \(k\to\infty\) is of size \((\frac{k}{2\pi})^d\) times a Duistermaat-Heckman integral independent of \(k\), where \(\dim\mathcal{O}_\lambda=2d\).
In the paper under review, the authors generalize this phenomenon to coadjoint orbits of central extensions of loop groups and of the diffeomorphism group of the circle. They show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contribution which can be interpreted as a formal Duistermaat-Heckman orbital integral.
Furthermore, reduced spaces of Virasoro coadjoint orbits are considered, and the authors suggest a new invariant which replaces symplectic volume in the infinite-dimensional situation. They also consider other modules of the Virasoro algebra (in particular, the modules corresponding to minimal models) and obtain Duistermaat-Heckman type expressions which do not correspond to any Virasoro coadjoint orbits.
Moreover, the authors introduce volume functions corresponding to formal Duistermaat-Heckman integrals over coadjoint orbits of the Virasoro algebra and show that they are related by the Hankel transform to the spectral densities recently studied by Saad, Shenker and Stanford.
Reviewer: Jan Frahm (Århus)Holomorphic Legendrian curves in \(\mathbb{CP}^3\) and superminimal surfaces in \(\mathbb{S}^4\)https://zbmath.org/1487.320552022-07-25T18:03:43.254055Z"Alarcón, Antonio"https://zbmath.org/authors/?q=ai:alarcon.antonio"Forstnerič, Franc"https://zbmath.org/authors/?q=ai:forstneric.franc"Lárusson, Finnur"https://zbmath.org/authors/?q=ai:larusson.finnurSummary: We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective \(3\)-space \(\mathbb{CP}^3\), both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into \(\mathbb{CP}^3\) is path-connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into \(\mathbb{CP}^3\) as a complete holomorphic Legendrian curve. Under the twistor projection \(\pi:\mathbb{CP}^3\to \mathbb{S}^4\) onto the \(4\)-sphere, immersed holomorphic Legendrian curves \(M\to \mathbb{CP}^3\) are in bijective correspondence with superminimal immersions \(M\to\mathbb{S}^4\) of positive spin, according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in \(\mathbb{S}^4\). In particular, superminimal immersions into \(\mathbb{S}^4\) satisfy the Runge approximation theorem and the Calabi-Yau property.Gromov-Witten invariants of \(\mathbb{P}^1\) coupled to a KdV tau functionhttps://zbmath.org/1487.320702022-07-25T18:03:43.254055Z"Norbury, Paul"https://zbmath.org/authors/?q=ai:norbury.paul-tSummary: We consider the pull-back of a natural sequence of cohomology classes \(\Theta_{g, n} \in H^{2 (2 g - 2+ n)}(\overline{\mathcal{M}}_{g, n}, \mathbb{Q})\) to the moduli space of stable maps \(\overline{\mathcal{M}}_{g, n}(\mathbb{P}^1, d)\). These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via \(Z^{B G W}(\hbar, t_0, t_1, \dots) = \exp \sum \frac{ \hbar^{2 g - 2}}{ n !} \int_{\overline{\mathcal{M}}_{g, n}} \Theta_{g, n} \cdot \prod_{j = 1}^n \psi_j^{k_j} \prod t_{k_j} \). Insertions of the pull-backs of the classes \(\Theta_{g, n}\) into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target \(\mathbb{P}^1\) are given by a random matrix integral and satisfy the Toda equation.The space of almost calibrated \((1,1)\)-forms on a compact Kähler manifoldhttps://zbmath.org/1487.320972022-07-25T18:03:43.254055Z"Chu, Jianchun"https://zbmath.org/authors/?q=ai:chu.jianchun"Collins, Tristan C."https://zbmath.org/authors/?q=ai:collins.tristan-c"Lee, Man-Chun"https://zbmath.org/authors/?q=ai:lee.man-chunSummary: The space \(\mathcal{H}\) of ``almost calibrated'' \((1,1)\)-forms on a compact Kähler manifold plays an important role in the study of the deformed Hermitian Yang-Mills equation of mirror symmetry, as emphasized by recent work of the second author et al. [in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 1. Oxford: Oxford University Press. 69--90 (2018; Zbl 1421.35300)], and is related by mirror symmetry to the space of positive Lagrangians studied by \textit{J. P. Solomon} [Math. Ann. 357, No. 4, 1389--1424 (2013; Zbl 1282.53067); Geom. Funct. Anal. 24, No. 2, 670--689 (2014; Zbl 1296.53157)]. This paper initiates the study of the geometry of \(\mathcal{H}\). We show that \(\mathcal{H}\) is an infinite-dimensional Riemannian manifold with nonpositive sectional curvature. In the hypercritical phase case we show that \(\mathcal{H}\) has a well-defined metric structure, and that its completion is a \(\mathrm{CAT}(0)\) geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case \(\mathcal{H}\) admits \(C^{1,1}\) geodesics, improving a result of [Zbl 1421.35300)]. Using results of \textit{T. Darvas} and \textit{L. Lempert} [Math. Res. Lett. 19, No. 5, 1127--1135 (2012; Zbl 1275.58008)] we show that this result is sharp.A symplectic form on the space of embedded symplectic surfaces and its reduction by reparametrizationshttps://zbmath.org/1487.321532022-07-25T18:03:43.254055Z"Kessler, Liat"https://zbmath.org/authors/?q=ai:kessler.liatSummary: Let \((M, \omega)\) be a symplectic manifold, and \((\Sigma, \sigma)\) a closed connected symplectic 2-manifold. We construct a weakly symplectic form \({\omega^D}\) on \(\operatorname{C}^{\infty}(\Sigma,M)\) which is a special case of Donaldson's form. We show that the restriction of~\({\omega^D}\) to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding \((\Sigma, \sigma) \hookrightarrow (M, \omega)\) descends to a weakly symplectic form on the quotient by \(\operatorname{Sympl}(\Sigma, \sigma)\), and that the symplectic space obtained is a symplectic quotient of the subspace of symplectic embeddings with respect\- to the \(\operatorname{Sympl}(\Sigma, \sigma)\)-action. We also compare \({\omega^D}\) to another 2-form. We conclude with a result on the restriction of \({\omega^D}\) to moduli spaces of holomorphic~curves.Almost existence from the feral perspective and some questionshttps://zbmath.org/1487.321542022-07-25T18:03:43.254055Z"Fish, Joel W."https://zbmath.org/authors/?q=ai:fish.joel-w"Hofer, Helmut H. W."https://zbmath.org/authors/?q=ai:hofer.helmut-h-wSummary: We use feral pseudoholomorphic curves and adiabatic degeneration to prove an extended version of the so-called `almost existence result' for regular compact Hamiltonian energy surfaces. That is, that for a variety of symplectic manifolds equipped with a Hamiltonian, almost every (non-empty) compact energy level has a periodic orbit.Caustics of weakly Lagrangian distributionshttps://zbmath.org/1487.350082022-07-25T18:03:43.254055Z"Gomes, Seán"https://zbmath.org/authors/?q=ai:gomes.sean-p"Wunsch, Jared"https://zbmath.org/authors/?q=ai:wunsch.jaredSummary: We study semiclassical sequences of distributions \(u_h\) associated with a Lagrangian submanifold of phase space \(\mathcal{L}\subset T^*X\). If \(u_h\) is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on \(\mathcal{L},\) then the asymptotics of \(u_h\) are well understood by work of Arnol'd, provided \(\mathcal{L}\) projects to \(X\) with a stable simple Lagrangian singularity. We establish sup-norm estimates on \(u_h\) under much more general hypotheses on the rate at which it is concentrating on \(\mathcal{L}\) (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.Wavelet packets associated with linear canonical transform on spectrumhttps://zbmath.org/1487.420802022-07-25T18:03:43.254055Z"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younus"Dar, Aamir H."https://zbmath.org/authors/?q=ai:dar.aamir-hWavelet packets generalize classical orthonormal wavelet bases by having better frequency localization. On the other hand, the linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms. Using LCT, the authors constructed wavelet packets corresponding to nonuniform multiresolution analysis associated with LCT and then those corresponding to vector-valued nonuniform multiresolution analysis associated with LCT. The results in this paper generalize various properties of classical wavelet packets by means of LCT.
Reviewer: Bin Han (Edmonton)Pseudo-differential operator in the framework of linear canonical transform domainhttps://zbmath.org/1487.460402022-07-25T18:03:43.254055Z"Prasad, Akhilesh"https://zbmath.org/authors/?q=ai:prasad.akhilesh"Ansari, Z. A."https://zbmath.org/authors/?q=ai:ansari.zahid-a|ansari.zamir-ahmad"Jain, Pankaj"https://zbmath.org/authors/?q=ai:jain.pankajA comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint. With an appendix by Igor Zelenkohttps://zbmath.org/1487.530012022-07-25T18:03:43.254055Z"Agrachev, Andrei"https://zbmath.org/authors/?q=ai:agrachev.andrej-a"Barilari, Davide"https://zbmath.org/authors/?q=ai:barilari.davide"Boscain, Ugo"https://zbmath.org/authors/?q=ai:boscain.ugoThis is a very well-written text that presents a thorough introduction to sub-Riemannian geometry. The results are well organized and carefully presented throughout the whole text. Actually, this textbook offers more than resources for specialists, it is also intended to students. It does a great presentation on more classical topics from differential and Riemannian geometry. The authors even give a possible selection of chapters that could serve as the basis of three possible graduate-level courses: one on Riemannian geometry; one at the introductory level on sub-Riemannian geometry; and one at an advanced level on sub-Riemannian geometry.
Sub-Riemannian geometry can be seen as a generalization of the Riemannian one, the main difference being that one ``cannot move, receive or send information in all directions''. Thus, in a nutshell, a sub-Riemannian structure on a smooth manifold has a fixed admissible subspace in any tangent space, which has some Euclidean setup. The admissible paths are those whose velocities are admissible, and these generate a suitable notion of distance between points.
These spaces turned out to be extremely rich in applications, and one obvious field for such is the optimal control theory. Therefore, the authors have a ``control theoretic'' mindset when presenting the results. In particular, they use the language of Hamiltonian dynamics throughout.
After an introduction, the book is divided into 21 chapters and an appendix. These chapters could be roughly categorized into three main parts. Chapters 1--5 represent Part one, these are presenting introductory concepts from the theory of surfaces, sub-Riemannian structures (such as the distance, characterization of geodesics, cf. Rashevskii-Chow theorem, Filippov's theorem and the Pontryagin maximum principle), symplectic geometry, Hamiltonian systems (as action-angle coordinates, geodesic flows, etc.).
Part 2 consists of Chapters 6--13. Some more classical results from functional analysis, operator calculus and Lie groups are presented in Chapters 6--7. The following chapter deals with notions such as the endpoint and exponential maps, cut and conjugate points. Chapter 9 presents in particular an interesting study on the Grushin plane. Chapter 10 discusses so-called nonholonomic tangent spaces, while Chapter 11 studies general analytic properties of the sub-Riemannian distance function. Chapter 12 turns to abnormal geodesics and Chapter 13 is devoted to some explicite calculations of the sub-Riemannian optimal synthesis for model spaces.
The remaining part from Chapter 14 to the end of the book can be seen as Part 3. These chapters are particularly devoted to the study of curvature notions and their applications (these include in particular the study of Jacobi curves; Levi-Civita connection; the Riemannian curvature and its symplectic meaning; Poisson manifolds; integrability of sub-Riemannian geodesic flows on 3D Lie groups, etc.). The last two chapters address the question of defining a canonical volume in sub-Riemannian geometry and the construction of the sub-Riemannian Laplace operator and the associated heat equation.
The theoretical results are accompanied by a good number of examples, by lots of suitable exercises and by bibliographical notes at the end of each chapter.
Overall, I find this text as an excellent resource for a broad range of topics in Riemannian and sub-Riemannian geometry. I am strongly convinced that this will become one of the main references for people interested in these topics, ranging from students to specialists.
Reviewer: Alpár R. Mészáros (Durham)Pseudo projective curvature tensor satisfying some properties on a normal paracontact metric manifoldhttps://zbmath.org/1487.530522022-07-25T18:03:43.254055Z"Yıldırım, Ümit"https://zbmath.org/authors/?q=ai:yildirim.umit"Atçeken, Mehmet"https://zbmath.org/authors/?q=ai:atceken.mehmet"Dirik, Süleyman"https://zbmath.org/authors/?q=ai:dirik.suleymanSummary: In the present paper we have studied the curvature tensor of a normal paracontact metric manifold satisfying the conditions \(R(\xi,X)\widetilde{P}=0\), \(\widetilde{P}(\xi,X)R=0\), \(\widetilde{P}(\xi,X)\widetilde{P}=0\), \(\widetilde{P}(\xi,X) S=0\), \(\widetilde{P}(\xi,X)\widetilde{Z}=0\) and pseudo projective flatness, where \(R\), \(\widetilde{P}\), \(S\) and \(\widetilde{Z}\) denote the Riemannian curvature, pseudo projective curvature, Ricci and concircular curvature tensors, respectively.Certain results on \(N(k)\)-contact metric manifolds and torse-forming vector fieldshttps://zbmath.org/1487.530532022-07-25T18:03:43.254055Z"Yoldaş, Halil İbrahim"https://zbmath.org/authors/?q=ai:yoldas.halil-ibrahimA contact metric manifold \(M\) is called \(N(k)\)-contact metric manifold if \(M\) is endowed with a \(k\)-nullity distribution. The author studies the properties of these manifolds endowed with a torse-forming vector field and admitting a Ricci soliton. Let us mention some of these results. The author finds conditions such that an \(N(k)\)-contact metric manifold becomes Sasakian. A characterization for a torse-forming vector field to be recurrent is provided. He finds a characterization in terms of non-zero potential vector field for a Ricci soliton to be shrinking.
Reviewer: Neda Bokan (Beograd)\(b\)-structures on Lie groups and Poisson reductionhttps://zbmath.org/1487.530942022-07-25T18:03:43.254055Z"Braddell, Roisin"https://zbmath.org/authors/?q=ai:braddell.roisin"Kiesenhofer, Anna"https://zbmath.org/authors/?q=ai:kiesenhofer.anna"Miranda, Eva"https://zbmath.org/authors/?q=ai:miranda.evaSummary: Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a \(b\)-Lie group as a pair \((G, H)\) where \(G\) is a Lie group and \(H\) is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical \(b\)-symplectic structure on the \(b\)-cotangent bundle \({}^b T^{\ast} G\) together with its reduction theory. Namely, we extend the minimal coupling procedure to \({}^b T^{\ast} G/H\) and prove that the Poisson reduction under the cotangent lifted action of \(H\) by left translations can be described in terms of the Lie Poisson structure on \(\mathfrak{h}^{\ast}\) (where \(\mathfrak{h}\) is the Lie algebra of \(H)\) and the canonical \(b\)-symplectic structure on \({}^b T^{\ast} (G/H)\), where \(G/H\) is viewed as a one-dimensional \(b\)-manifold having as critical hypersurface (in the sense of \(b\)-manifolds) the identity element.Non-invariant hypersurfaces of A \(( \epsilon,\delta )\)-trans Sasakian manifoldshttps://zbmath.org/1487.530952022-07-25T18:03:43.254055Z"Khan, Toukeer"https://zbmath.org/authors/?q=ai:khan.toukeer"Rizvi, Sheeba"https://zbmath.org/authors/?q=ai:rizvi.sheebaSummary: The object of this paper is to study non-invariant hypersurface of a \((\epsilon,\delta)\)-trans Sasakian manifolds equipped with \((f,g,u,v,\lambda )\)-structure. Some properties obeyed by this structure are obtained. The necessary and sufficient conditions also have been obtained for totally umbilical non-invariant hypersurface with \((f,g,u,v,\lambda )\)-structure of a \(( \epsilon, \delta )\)-trans Sasakian manifolds to be totally geodesic. The second fundamental form of a non-invariant hypersurface of a \(( \epsilon, \delta )\)-trans Sasakian manifolds with \((f,g,u,v,\lambda )\)-structure has been traced under the condition when \(f\) is parallel.Twistor operators in symplectic geometryhttps://zbmath.org/1487.530962022-07-25T18:03:43.254055Z"Krýsl, Svatopluk"https://zbmath.org/authors/?q=ai:krysl.svatoplukSummary: On a symplectic manifold equipped with a symplectic connection and a metaplectic structure, we define two families of sequences of differential operators, called the symplectic twistor operators. We prove that if the connection is torsion-free and Weyl-flat, the sequences in these families form complexes.On 2-plectic Lie groupshttps://zbmath.org/1487.530972022-07-25T18:03:43.254055Z"Shafiee, Mohammad"https://zbmath.org/authors/?q=ai:shafiee.mohammad-ali"Aminizadeh, Masoud"https://zbmath.org/authors/?q=ai:aminizadeh.masoudSummary: A 2-plectic Lie group is a Lie group endowed with a 2-plectic structure which is left invariant. In this paper we provide some interesting examples of 2-plectic Lie groups. Also we study the structure of the set of Hamiltonian covectors and vectors of a 2-plectic Lie algebra. Moreover, the existence of \(i\)-isotropic and \(i\)-Lagrangian subgroups are investigated. At last we obtain some results about the reduction of some 2-plectic structures.A new class of contact pseudo framed manifolds with applicationshttps://zbmath.org/1487.530982022-07-25T18:03:43.254055Z"Duggal, K. L."https://zbmath.org/authors/?q=ai:duggal.krishan-lalOn a given manifold \(M\) fix the data \((g, \eta, \xi, f, \lambda)\) with \(g\) a semi-Riemannian metric, \(\eta \) a \(1\)-form, \(\xi \) a vector field which is not null and \(\lambda \) a smooth nonzero function. The author calls \((M, g, \eta, \xi, f, \lambda )\) an almost metric contact pseudo framed (CPF) manifold if \(\eta (\xi )=1\), \(f^3=\lambda ^2f\), \(g(fX, fY)=\mu (g(X, Y)-\varepsilon \sigma ^2\eta (X)\eta (Y))\), \(g(\xi, \xi )=\varepsilon \sigma ^2\), \(g(X, \xi )=\varepsilon \sigma ^2\eta (X)\) where \(\varepsilon =+1\) or \(-1\) if \(\xi \) is space-like or time-like and \(\sigma , \mu \) are other two nonzero smooth functions. Several examples and various types of symmetries, including Ricci solitons, are discussed, some of them inspired by physical models. Two open problems are raised.
Reviewer: Mircea Crâşmăreanu (Iaşi)Gradient generalized \(\eta\)-Ricci soliton and contact geometryhttps://zbmath.org/1487.530992022-07-25T18:03:43.254055Z"Ghosh, Amalendu"https://zbmath.org/authors/?q=ai:ghosh.amalenduSummary: In this paper, we consider gradient generalized \(\eta\)-Ricci soliton on contact metric manifolds of dimension \(\geq 5\). First, we prove that if a \(K\)-contact metric represents a gradient generalized \(\eta\)-Ricci soliton then either it is compact \(\eta\)-Einstein and Sasakian, provided \(\lambda > -2\), or it is compact and isometric to a unit sphere \(S^{2n+1} (1)\). Next we prove that, if a compact contact metric with parallel Ricci tensor represents a non-trivial gradient generalized \(\eta\)-Ricci soliton, then it is locally isometric to \(S^{2n+1}(1)\).Symplectic diffeomorphisms and Weinstein 1-formhttps://zbmath.org/1487.531002022-07-25T18:03:43.254055Z"Luganda, Fidele Balibuno"https://zbmath.org/authors/?q=ai:luganda.fidele-balibuno"Todjihounde, Leonard"https://zbmath.org/authors/?q=ai:todjihounde.leonardSummary: In [Indiana Univ. Math. J. 22, 267--275 (1972; Zbl 0237.58002)] \textit{J. Sniatycki} and \textit{W. M. Tulczyjew} showed that the Liouville 1-form lying on the cotangent bundle is derived from physical potential and is related to the symplectomorphism through the flux homomorphism. On the other hand, in [Adv. Math. 6, 329--346 (1971; Zbl 0213.48203); Ann. Math. (2) 98, 377--410 (1973; Zbl 0271.58008)], \textit{A. Weinstein} constructed a chart from the group of symplectic diffeomorphisms isotopic to the identity by using Lagrangian sub-manifolds geometry and from which he derived a closed 1-form called the Weinstein 1-form. In this paper, we establish a relation between the Liouville 1-form and the Weinstein 1-form through an explicit formula from which we derive a new characterization of symplectomorphism and a new formula of the flux homomorphism.On a type of \(\alpha\)-cosymplectic manifoldshttps://zbmath.org/1487.531012022-07-25T18:03:43.254055Z"Beyendi, Selahattin"https://zbmath.org/authors/?q=ai:beyendi.selahattin"Ayar, Gühan"https://zbmath.org/authors/?q=ai:ayar.guhan"Aktan, Nesip"https://zbmath.org/authors/?q=ai:aktan.nesipSummary: The object of this paper is to study \(\alpha\)-cosymplectic manifolds admitting a \(W_2\)-curvature tensor.On almost \(\alpha\)-para-Kenmotsu manifolds satisfying certain conditionshttps://zbmath.org/1487.531022022-07-25T18:03:43.254055Z"Erken, Irem Küpeli"https://zbmath.org/authors/?q=ai:erken.irem-kupeliSummary: In this paper, we study some remarkable properties of almost \(\alpha \)-para-Kenmotsu manifolds. We consider projectively flat, conformally flat and concircularly flat almost \(\alpha\)-para-Kenmotsu manifolds (with the \(\eta\)-parallel tensor field \(\phi h\)). Finally, we present an example to verify our results.Loxodromes and transformations in pseudo-Hermitian geometryhttps://zbmath.org/1487.531032022-07-25T18:03:43.254055Z"Lee, Ji-Eun"https://zbmath.org/authors/?q=ai:lee.jieunSummary: In this paper, we prove that a diffeomorphism \(f\) on a normal almost contact \(3\)-manifold \(M\) is a CRL-\textit{transformation} if and only if \(M\) is an \(\alpha\)-Sasakian manifold. Moreover, we show that a \(CR\)-loxodrome in an \(\alpha\)-Sasakian \(3\)-manifold is a pseudo-Hermitian magnetic curve with a strength \(q=\widetilde{r}\eta (\gamma^{\prime})=(r+\alpha -t)\eta (\gamma^{\prime})\) for constant \(\eta (\gamma^{\prime})\). A non-geodesic \(CR\)-loxodrome is a non-Legendre slant helix. Next, we prove that let \(M\) be an \(\alpha\)-Sasakian \(3\)-manifold such that \((\nabla_Y S)X=0\) for vector fields \(Y\) to be orthogonal to \(\xi\), then the Ricci tensor \(\rho\) satisfies \(\rho =2\alpha^2 g\). Moreover, using the CRL-\textit{transformation} \(\widetilde{\nabla}^t\) we fine the pseudo-Hermitian curvature \(\widetilde{R}\), the pseudo-Ricci tensor \(\widetilde{\rho}\) and the torsion tensor field \(\widetilde{\mathfrak{T}}^t (\widetilde{S}X,Y)\).Quasi-Sasakian structures on 5-dimensional nilpotent Lie algebrashttps://zbmath.org/1487.531042022-07-25T18:03:43.254055Z"Özdemir, Nülifer"https://zbmath.org/authors/?q=ai:ozdemir.nulifer"Aktay, Şirin"https://zbmath.org/authors/?q=ai:aktay.sirin"Solgun, Mehmet"https://zbmath.org/authors/?q=ai:solgun.mehmetSummary: In this study, we examine the existence of quasi-Sasakian structures on nilpotent Lie algebras of dimension five. In addition, we give some results about left invariant quasi-Sasakian structures on Lie groups of dimension five, whose Lie algebras are nilpotent. Moreover, subclasses of quasi-Sasakian structures are studied for some certain classes.On \(\phi\)-symmetric \(N(k)\)-paracontact metric manifoldshttps://zbmath.org/1487.531052022-07-25T18:03:43.254055Z"Prakasha, D. G."https://zbmath.org/authors/?q=ai:prakasha.doddabhadrappla-gowda"Mirji, K. K."https://zbmath.org/authors/?q=ai:mirji.k-kSummary: The notions of \(\phi\)-symmetric, 3-dimensional locally \(\phi\)-symmetric, \(\phi\)-Ricci symmetric, and 3-dimensional locally \(\phi\)-Ricci symmetric \(N(k)\)-paracontact metric manifolds have been introduced and properties of these structures have been discussed.Vertical and complete lifts of sections of a (dual) vector bundle and Legendre dualityhttps://zbmath.org/1487.531062022-07-25T18:03:43.254055Z"Peyghan, E."https://zbmath.org/authors/?q=ai:peyghan.esmaeil"Nourmohammadifar, L."https://zbmath.org/authors/?q=ai:nourmohammadifar.leila"Arcuş, C. M."https://zbmath.org/authors/?q=ai:arcus.constantin-mSummary: Using the covariant derivative for exterior forms of a (dual) vector bundle, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical lifts and between complete lifts are presented. Finally, a duality between Lie algebroids structures is developed.Symplectic and Kähler structures on \(\mathbb{C}P^1\)-bundles over \(\mathbb{C}P^2\)https://zbmath.org/1487.531072022-07-25T18:03:43.254055Z"Lindsay, Nicholas"https://zbmath.org/authors/?q=ai:lindsay.nicholas"Panov, Dmitri"https://zbmath.org/authors/?q=ai:panov.dmitriThe authors focus on Tolman's manifolds to study symplectic manifolds which does not admit Kähler metric.
Tolman's manifolds are compact symplectic manifolds of dimension 6 with Hamiltonian torus actions constructed by \textit{S. Tolman}
[Invent. Math. 131, no. 2, 299--310 (1998; Zbl 0901.58018)].
The authors discuss when the symplectic manifolds does not admit Kähler structures compatible with them.
As a result, they describe some conditions for such a Kähler metric to exist.
The paper consists of five sections. Section 1 is a commentary on the background of the study. Main theorems (Theorem 1.3, 1.5 and 1.6) are also provided in the section.
Section 2 is the quick review of the construction of Tolman's manifolds. The authors explain simply some results on Tolman's manifolds
including Hamiltonian circle actions. Section 3 treats topological aspects of Tolman's manifold needed for proving Theorem 1.3.
The authors describe the intersection form on the second cohomology group of Tolman's manifold \(M_{\mathcal{T}}\) with the Chern classes to prove that
\(M_\mathcal{T}\) is diffeomorphic to the projectivisation \(\mathbb{P}(E)\) of a complex two bundle \(E\) over the complex projective space \(\mathbb{C}P^2\).
The proofs of main theorems begin from Section 4.
In Section 4, the authors prove the main theorem 1.3 and 1.6. Theorem 1.6 is shown first
by using basic properties of holomorphic rank two bundles over \(\mathbb{C}P^2\).
The proof of Theorem 1.3 uses Mori theory together with classical results on holomorphic rank two bundles over \(\mathbb{C}P^2\).
Section 5 is devoted to the proof of Theorem 1.5. The basic results on projective manifolds with circle actions is reviewed plain and the theorem is shown by using them.
Reviewer: Yuji Hirota (Sagamihara)Symplectic toric geometry and the regular dodecahedronhttps://zbmath.org/1487.531082022-07-25T18:03:43.254055Z"Prato, Elisa"https://zbmath.org/authors/?q=ai:prato.elisaSummary: The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.The geodesic flow on nilpotent Lie groups of steps two and threehttps://zbmath.org/1487.531092022-07-25T18:03:43.254055Z"Ovando, Gabriela P."https://zbmath.org/authors/?q=ai:ovando.gabriela-pThis work investigates the geodesic flow on \(k\)-step nilpotent Lie groups for \(k=2,3\). For such Lie groups in dimension at most five, the author obtains a left-invariant metric for which the geodesic flow is Liouville integrable. The main tools used to obtain Liouville integrability are Killing vector fields and symmetric Killing 2-tensors associated to quadratic invariant first integrals of the geodesic flow. The author obtains explicit conditions for a symmetric map to induce a first integral as well as conditions for invariant linear or quadratic polynomials to become first integrals. Several explicit involution formulas for first integrals are obtained. Moreover, they also investigate six-dimensional \(k\)-step nilpotent Lie groups for \(k=2,3\), and in most cases (all but two) show that these groups admit a left-invariant metric for which the geodesic flow is completely integrable.
Reviewer: Julie Rowlett (Gothenburg)Meromorphic connections, determinant line bundles and the Tyurin parametrizationhttps://zbmath.org/1487.531102022-07-25T18:03:43.254055Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Hurtubise, Jacques"https://zbmath.org/authors/?q=ai:hurtubise.jacques-cSummary: We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the ``sheaf of holomorphic connections'' (the sheaf of holomorphic splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, are rather quite general, for example, they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising from ``Tyurin families'' of stable bundles over the surface. These families generalize the Tyurin parametrization of stable vector bundles \(E\) over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) holomorphic connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of holomorphic connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.Symplectic \((-2)\)-spheres and the symplectomorphism group of small rational 4-manifolds. IIhttps://zbmath.org/1487.531112022-07-25T18:03:43.254055Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.7|li.jun.12|li.jun.14|li.jun.10|li.jun.2|li.jun.6|li.jun.13|li.jun|li.jun.8|li.jun.3|li.jun.1|li.jun.11"Li, Tian-Jun"https://zbmath.org/authors/?q=ai:li.tian-jun|li.tianjun"Wu, Weiwei"https://zbmath.org/authors/?q=ai:wu.weiwei.1|wu.weiwei|wu.weiwei.2Authors' abstract: We study the symplectic mapping class groups of \((\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}},\omega)\). Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures \(J_0\) as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by \textit{W. Wu} [Math. Ann. 359, No. 1--2, 153--168 (2014; Zbl 1315.53099)]) and the study of a semi-toric model to understand a ``connecting map'', whose cokernel is the symplectic mapping class group.
Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms \(\omega\). Let \(N_\omega\) the number of \((-2)\)-symplectic spherical homology classes. Torelli SMCG is trivial if \(N_\omega>8\); it is \(\pi_0(Diff^+(S^2,5))\) if \(N_{\omega}=0\) (by \textit{P. Seidel} [Lect. Notes Math. 1938, 231--267 (2008; Zbl 1152.53069)] and \textit{J. D. Evans} [J. Symplectic Geom. 9, No. 1, 45--82 (2011; Zbl 1242.58004)]); and it is \(\pi_0(\mathrm{Diff}^+(S^2,4))\) in the remaining case. Further, we completely determine the rank of \(\pi_1(\mathrm{Symp}(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}}, \omega)\) for any given symplectic form. Our results can be uniformly presented in terms of Dynkin diagrams of type-A and type-\(\mathbb{D}\) Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds (open problem 16 in \textit{D. McDuff} and \textit{D. Salamon}'s book [Introduction to symplectic topology. 3rd edition. Oxford: Oxford University Press (2016; Zbl 1380.53003)]).
Reviewer: Nicolai K. Smolentsev (Kemerovo)Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structureshttps://zbmath.org/1487.531122022-07-25T18:03:43.254055Z"Bao, Erkao"https://zbmath.org/authors/?q=ai:bao.erkao"Honda, Ko"https://zbmath.org/authors/?q=ai:honda.koThe authors define equivariant Langrangian Floer cohomology for an equivariant almost complex structure that is not necessarily regular by constructing equivariant semi-global Kuranishi structures similar to those used in [the authors, ``Semi-global Kuranishi charts and the definition of contact homology'', Preprint, \url{arXiv:1512.00580}].
Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) that is either closed or has contact type boundary, and assume that \(L_0\) and \(L_1\) are oriented Lagrangian submanifolds that intersect transversally. Let \(G\) be a group that acts symplectically on \((M,\omega)\) such that \(g(L_0) = L_0\) and \(g(L_1) = L_1\) for all \(g \in G\), and assume that \(G\) fixes the orientations on \(L_0\) and \(L_1\). Let \(J\) be a \(G\)-invariant and \(\omega\)-compatible almost complex structure on \(M\). The authors make several standard assumptions such as
(S) The maps \(\pi_2(M) \stackrel{\int \omega}{\rightarrow} \mathbb{R}\) and \(\pi_2(M,L_i) \stackrel{\int \omega}{\rightarrow} \mathbb{R}\) for \(i=0,1\) have image 0.
(O) The pair \((L_0,L_1)\) is equipped with a relative spin structure which is preserved by \(G\).
(J) an assumption concerning the compatibility of the almost complex structure \(J\) with the contact form and Reeb vector field on a collar neighborhood of the boundary when \(\partial M \neq \emptyset\).
The chain groups in the Lagrangian Floer cochain complex \(CF^\bullet(L_0,L_1)\) are free modules over a Novikov ring \(R\) generated by points in the intersection \(L_0 \cap L_1\), and the boundary operator is defined by counting \(J\)-holomorphic strips between intersection points \(p,q \in L_0 \cap L_1\). This count is given by the number of elements in a moduli space \(\mathcal{M}_J(p,q;A)\), where \(A\) represents a homotopy class of continuous strips connecting \(p\) and \(q\).
In general, a \(G\)-invariant almost complex structure \(J\) won't be regular, and hence certain transversality conditions commonly used to prove that \(\mathcal{M}_J(p,q;A)\) is a manifold won't hold. In the words of the authors, ``The main contribution of this paper is to obtain a \(G\)-equivariant cochain complex \(CF^\bullet(L_0,L_1)\) when \(J\) is not regular by constructing an equivariant version of a semi-global Kuranishi structure, initially developed in [\textit{E. Bao} and \textit{K. Honda}, ``Semi-global Kuranishi charts and the definition of contact homology'', Preprint, \url{arXiv:1512.00580}] for contact homology''.
The bulk of the paper consists of detailed constructions of equivariant semi-global Kuranishi structures used to define boundary maps, chain maps, and chain homotopies for equivariant Lagrangian Floer cohomology. For instance, to define the boundary operator the authors replace the moduli space \(\mathcal{M}_J(p,q;A)\), which might not be a manifold, with a space \(\mathcal{Z}(\mathscr{K}(p,q;A), \mathfrak{G})\) defined by the Kuranishi structure which is a manifold by Lemma 2.7.3 of the paper.
The constructions of the required Kuranishi structures imply the following main theorem:
\textbf{Theorem 1.0.3} (equivariant Lagrangian Floer cohomology) Suppose \(G\) acts on \((M,\omega)\) symplectically and, for each \(i=0,1\), \(L_i\) is oriented, \(g_i(L_i) = L_i\) for each \(g \in G\), and \(G\) fixes the orientation of \(L_i\). If (S) and (O) hold, then there exists an \(R\)-module \(HF_G^\bullet(L_0, L_1)\) which is an invariant of \((L_0,L_1)\) under \(G\)-equivariant Hamiltonian isotopy. Moreover, when there exists a regular \(G\)-invariant \(\omega\)-compatible almost complex structure on \(M\) satisfying (J), the usual definition of equivariant Lagrangian Floer cohomology can be made and agrees with \(HF_G^\bullet(L_0, L_1)\).
Reviewer: David E. Hurtubise (Altoona)Deformations and representations of Lie algebroidshttps://zbmath.org/1487.580122022-07-25T18:03:43.254055Z"Bouaziz, Emile"https://zbmath.org/authors/?q=ai:bouaziz.emileSummary: We study a class of derived representations of a Lie algebroid. The dg-category of these representations enhances the classical category of representations in the sense that the cohomology objects of a derived representation are classical representations. The adjoint complex is canonically an object of our category of representations. Our main contribution is the construction of an extension of the functor of de Rham-Lie cochains to this category, referred to as Crainic-Moerdijk-cochains. We show that, when applied to the adjoint representation, we obtain the Deformation Complex of Crainic and Moerdijk.Sub-Riemannian limit of the differential form heat kernels of contact manifoldshttps://zbmath.org/1487.580202022-07-25T18:03:43.254055Z"Albin, Pierre"https://zbmath.org/authors/?q=ai:albin.pierre"Quan, Hadrian"https://zbmath.org/authors/?q=ai:quan.hadrianAuthors' abstract: We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the \(\eta\)-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative \(\eta\)-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.
Reviewer: Mohammed El Aïdi (Bogotá)Berezin-Toeplitz quantization associated with higher Landau levels of the Bochner Laplacianhttps://zbmath.org/1487.580242022-07-25T18:03:43.254055Z"Kordyukov, Yuri A."https://zbmath.org/authors/?q=ai:kordyukov.yuri-aIn his landmark paper, \textit{F. A. Berezin} [Commun. Math. Phys. 40, 153--174 (1975; Zbl 1272.53082)] introduced a rigorous mathematical approach to the quantization question, that led, due to the vast landscape to be approached, to multifarious generalizations under quite different perspectives; for some more recent review articles, see e.g., [\textit{M. Schlichenmaier}, Adv. Math. Phys. 2010, Article ID 927280, 38 p. (2010; Zbl 1207.81049); \textit{M. Engliš}, Oper. Theory: Adv. Appl. 251, 69--115 (2016; Zbl 1341.53122); \textit{S. T. Ali} and \textit{M. Engliš}, Rev. Math. Phys. 17, No. 4, 391--490 (2005; Zbl 1075.81038); \textit{X. Ma}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 785--810 (2011; Zbl 1229.53088)]. In this paper, the author presents a Toeplitz operator calculus leading to the construction of a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, he chooses a Riemannian metric on the manifold such that the associated Bochner Laplacian, under a certain condition on the metric, has the same local model at each point. The technically elaborated procedure leads to a more general quantization than in the almost-Kähler case.
Reviewer: Antonio Roberto da Silva (Rio de Janeiro)Semiclassical spectral analysis of the Bochner-Schrödinger operator on symplectic manifolds of bounded geometryhttps://zbmath.org/1487.580292022-07-25T18:03:43.254055Z"Kordyukov, Yuri A."https://zbmath.org/authors/?q=ai:kordyukov.yuri-aLet \((L,h^L)\) be a Hermitian line bundle with bounded geometry on \(X\), a smooth Riemannian manifold of bounded geometry endowed with a Hermitian connection \(\nabla^L\) and \((E,h^E)\) be a Hermitian vector bundle of rank \(k\) with bounded geometry on \(X\) with a Hermitian connection \(\nabla^E\). Let \(p\in\mathbb N\), we denote by \(L_p\) the \(pth\) tensor power of \(L\) and \(\nabla^{L_p\otimes E}\) be the \(C^\infty(X,T^*X\otimes L_p\otimes E)\)-valued Hermitian connection on \(C^\infty(X,L_p\otimes E)\). The induced Bochner Laplacian operator is defined by \(\Delta^{L_p\otimes E}=(\nabla^{L_p\otimes E})^*\nabla^{L_p\otimes E}\) where \((\nabla^{L_p\otimes E})^*\) is the formal adjoint associated to \(\nabla^{L_p\otimes E}\). Let \(V\) be a self-adjoint endomorphism of \(E\). The purpose of the author is to study properties of the operator \(H_p=\frac{1}{p}\Delta^{\nabla^{L_p\otimes E}}+V\) on \(C^\infty(X,L_p\otimes E)\). E.g., he provides an asymptotic description of the spectrum of \(H_p\) (as \(p\) goes to the infinity) in terms of the spectra of the model operators, i.e., some suitable second order differential operators obtained from \(H_p\).
Reviewer: Mohammed El Aïdi (Bogotá)A \(K\)-contact Lagrangian formulation for nonconservative field theorieshttps://zbmath.org/1487.700952022-07-25T18:03:43.254055Z"Gaset, Jordi"https://zbmath.org/authors/?q=ai:gaset.jordi"Gràcia, Xavier"https://zbmath.org/authors/?q=ai:gracia.xavier"Muñoz-Lecanda, Miguel C."https://zbmath.org/authors/?q=ai:munoz-lecanda.miguel-c"Rivas, Xavier"https://zbmath.org/authors/?q=ai:rivas.xavier"Román-Roy, Narciso"https://zbmath.org/authors/?q=ai:roman-roy.narcisoSummary: Dynamical systems with dissipative behaviour can be described in terms of contact manifolds and a modified version of Hamilton's equations. Dissipation terms can also be added to field equations, as showed in a recent paper where we introduced the notion of \(k\)-contact structure, and obtained a modified version of the De Donder-Weyl equations of covariant Hamiltonian field theory. In this paper we continue this study by presenting a \(k\)-contact Lagrangian formulation for nonconservative field theories. The Lagrangian density is defined on the product of the space of \(k\)-velocities times a \(k\)-dimensional Euclidean space with coordinates \(s^\alpha\), which are responsible for the dissipation. We analyze the regularity of such Lagrangians; only in the regular case we obtain a \(k\)-contact Hamiltonian system. We study several types of symmetries for \(k\)-contact Lagrangian systems, and relate them with dissipation laws, which are analogous to conservation laws of conservative systems. Several examples are discussed: we find contact Lagrangians for some kinds of second-order linear partial differential equations, with the damped membrane as a particular example, and we also study a vibrating string with a magnetic-like term.Morse potential in relativistic contexts from generalized momentum operator: Schottky anomalies, Pekeris approximation and mappinghttps://zbmath.org/1487.810842022-07-25T18:03:43.254055Z"Gomez, Ignacio S."https://zbmath.org/authors/?q=ai:gomez.ignacio-s"Santos, Esdras S."https://zbmath.org/authors/?q=ai:santos.esdras-s"Abla, Olavo"https://zbmath.org/authors/?q=ai:abla.olavoNon-commutativity and non-inertial effects on a scalar field in a cosmic string space-time. I: Klein-Gordon oscillatorhttps://zbmath.org/1487.830812022-07-25T18:03:43.254055Z"Cuzinatto, Rodrigo Rocha"https://zbmath.org/authors/?q=ai:cuzinatto.rodrigo-rocha"de Montigny, Marc"https://zbmath.org/authors/?q=ai:de-montigny.marc"Pompeia, Pedro José"https://zbmath.org/authors/?q=ai:pompeia.pedro-joseWeak-field limit of \(f(R)\) gravity to unify peculiar white dwarfshttps://zbmath.org/1487.850222022-07-25T18:03:43.254055Z"Kalita, Surajit"https://zbmath.org/authors/?q=ai:kalita.surajit"Sarmah, Lupamudra"https://zbmath.org/authors/?q=ai:sarmah.lupamudraSummary: In recent years, the idea of sub- and super-Chandrasekhar limiting mass white dwarfs (WDs), which are potential candidates to produce under- and over-luminous type Ia supernovae, respectively, has been a key interest in the scientific community. Although researchers have proposed different models to explain these peculiar objects, modified theories of Einstein's gravity, particularly \(f(R)\) gravity with \(R\) being the scalar curvature, seem to be one of the finest choices to explain both the regimes of these peculiar WDs. It was already shown that considering higher-order corrections to the Starobinsky model with two parameters, the structure of sub- and super-Chandrasekhar progenitor WDs can be explained self consistently. It is also well-known that WDs can be considered Newtonian objects because of their large size. In this paper, we derive the weak-field limit of \(f(R)\) gravity, which turns out to be the higher-order correction to the Poisson equation. Later, we use this equation to obtain the structures of sub- and super-Chandrasekhar limiting mass WDs at various central densities incorporating just one model parameter.