In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an begin{document}$ n $end{document} -dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an begin{document}$ (n-1) $end{document} -dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.
In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an begin{document}$ n $end{document} -dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an begin{document}$ (n-1) $end{document} -dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.
{"title":"A family of multiply warped product semi-Riemannian Einstein metrics","authors":"B. Pal, Pankaj Kumar","doi":"10.3934/jgm.2020017","DOIUrl":"https://doi.org/10.3934/jgm.2020017","url":null,"abstract":"In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an begin{document}$ n $end{document} -dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an begin{document}$ (n-1) $end{document} -dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86039280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There is an error in the statement of Theorem 4.25 in [1], a somewhat related typographical error in Remark 4.26, and an error in Remark 4.27 following directly from that in Theorem 4.25. Footnote 8 is also now obsolete. In order to ensure that the errors are unambiguously fixed, what appears below should replace the original text starting from just before the statement of Theorem 4.25 and ending at the end of Section 4.
{"title":"Erratum for 'nonholonomic and constrained variational mechanics'","authors":"A. D. Lewis","doi":"10.3934/jgm.2020033","DOIUrl":"https://doi.org/10.3934/jgm.2020033","url":null,"abstract":"There is an error in the statement of Theorem 4.25 in [1], a somewhat related typographical error in Remark 4.26, and an error in Remark 4.27 following directly from that in Theorem 4.25. Footnote 8 is also now obsolete. In order to ensure that the errors are unambiguously fixed, what appears below should replace the original text starting from just before the statement of Theorem 4.25 and ending at the end of Section 4.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87041492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamiltonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin-quadrupole Hamiltonians of Mead [Phys. Rev. Lett. 59, 161–164 (1987)] and Avron et al [Commun. Math. Phys. 124(4), 595–627 (1989)]. An explicit correspondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues.
{"title":"Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries","authors":"T. Iwai, D. Sadovskií, B. Zhilinskií","doi":"10.3934/jgm.2020021","DOIUrl":"https://doi.org/10.3934/jgm.2020021","url":null,"abstract":"We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamiltonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin-quadrupole Hamiltonians of Mead [Phys. Rev. Lett. 59, 161–164 (1987)] and Avron et al [Commun. Math. Phys. 124(4), 595–627 (1989)]. An explicit correspondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88468258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.
我们用几何工具研究了丝状结构的接触和化简，定义了李群上的哈密顿结构和动量映射。
{"title":"Invariant structures on Lie groups","authors":"J. P. Álvarez","doi":"10.3934/jgm.2020007","DOIUrl":"https://doi.org/10.3934/jgm.2020007","url":null,"abstract":"We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89244001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Avendaño-Camacho, Isaac Hasse-Armengol, E. Velasco-Barreras, Y. Vorobiev
On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.
{"title":"The method of averaging for Poisson connections on foliations and its applications","authors":"M. Avendaño-Camacho, Isaac Hasse-Armengol, E. Velasco-Barreras, Y. Vorobiev","doi":"10.3934/jgm.2020015","DOIUrl":"https://doi.org/10.3934/jgm.2020015","url":null,"abstract":"On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78290931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.
{"title":"Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy","authors":"M. Rodríguez-Olmos","doi":"10.3934/jgm.2020019","DOIUrl":"https://doi.org/10.3934/jgm.2020019","url":null,"abstract":"We survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78727675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a family of birational maps begin{document}$ varphi_k $end{document} in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family begin{document}$ varphi_k $end{document} using Poisson geometry tools, namely the properties of the restrictions of the maps begin{document}$ varphi_k $end{document} and their fourth iterate begin{document}$ varphi^{(4)}_k $end{document} to the symplectic leaves of an appropriate Poisson manifold begin{document}$ (mathbb{R}^4_+, P) $end{document} . These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product begin{document}$ SL(2, mathbb{Z})ltimesmathbb{R}^2 $end{document} . The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for begin{document}$ varphi_k $end{document} characterized by the parameter values begin{document}$ k = 1 $end{document} , begin{document}$ k = 2 $end{document} and begin{document}$ kgeq 3 $end{document} .
We consider a family of birational maps begin{document}$ varphi_k $end{document} in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family begin{document}$ varphi_k $end{document} using Poisson geometry tools, namely the properties of the restrictions of the maps begin{document}$ varphi_k $end{document} and their fourth iterate begin{document}$ varphi^{(4)}_k $end{document} to the symplectic leaves of an appropriate Poisson manifold begin{document}$ (mathbb{R}^4_+, P) $end{document} . These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product begin{document}$ SL(2, mathbb{Z})ltimesmathbb{R}^2 $end{document} . The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for begin{document}$ varphi_k $end{document} characterized by the parameter values begin{document}$ k = 1 $end{document} , begin{document}$ k = 2 $end{document} and begin{document}$ kgeq 3 $end{document} .
{"title":"The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps","authors":"I. Cruz, H. Mena-Matos, Esmeralda Sousa-Dias","doi":"10.3934/jgm.2020010","DOIUrl":"https://doi.org/10.3934/jgm.2020010","url":null,"abstract":"We consider a family of birational maps begin{document}$ varphi_k $end{document} in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family begin{document}$ varphi_k $end{document} using Poisson geometry tools, namely the properties of the restrictions of the maps begin{document}$ varphi_k $end{document} and their fourth iterate begin{document}$ varphi^{(4)}_k $end{document} to the symplectic leaves of an appropriate Poisson manifold begin{document}$ (mathbb{R}^4_+, P) $end{document} . These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product begin{document}$ SL(2, mathbb{Z})ltimesmathbb{R}^2 $end{document} . The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for begin{document}$ varphi_k $end{document} characterized by the parameter values begin{document}$ k = 1 $end{document} , begin{document}$ k = 2 $end{document} and begin{document}$ kgeq 3 $end{document} .","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76564243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a toric generalised Kähler structure on $ mathbb{C}P^2 $ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.
{"title":"Generalised Kähler structure on $ mathbb{C}P^2 $ and elliptic functions","authors":"F. Bonechi, J. Qiu, M. Tarlini","doi":"10.3934/jgm.2023009","DOIUrl":"https://doi.org/10.3934/jgm.2023009","url":null,"abstract":"We construct a toric generalised Kähler structure on $ mathbb{C}P^2 $ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76395797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$T$ map where $T in {mathbb R}^n$ is the period of the acting $n$-torus.
我们将具有有效哈密顿环面作用的完全可积哈密顿系统描述为具有零传输成本的系统。时间- T -映射，其中T in {mathbb R}^n$是作用的n$环面的周期。
{"title":"Characterization of toric systems via transport costs","authors":"Sonja Hohloch","doi":"10.3934/jgm.2020027","DOIUrl":"https://doi.org/10.3934/jgm.2020027","url":null,"abstract":"We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$T$ map where $T in {mathbb R}^n$ is the period of the acting $n$-torus.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74341204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerns an intrinsic formulation of nonholonomic mechanics. Our point of departure is the paper [ 6 ], by Koiller et al., revisiting E. Cartan's address at the International Congress of Mathematics held in 1928 at Bologna, Italy ([ 3 ]). Two notions of equivalence for nonholonomic mechanical systems begin{document}$ ( {mathsf{{M}}}, {{mathsf{{g}}}}, {mathscr{D}}) $end{document} are introduced and studied. According to [ 6 ], the notions of equivalence considered in this paper coincide. A counterexample is presented here showing that this coincidence is not always true.
This paper concerns an intrinsic formulation of nonholonomic mechanics. Our point of departure is the paper [ 6 ], by Koiller et al., revisiting E. Cartan's address at the International Congress of Mathematics held in 1928 at Bologna, Italy ([ 3 ]). Two notions of equivalence for nonholonomic mechanical systems begin{document}$ ( {mathsf{{M}}}, {{mathsf{{g}}}}, {mathscr{D}}) $end{document} are introduced and studied. According to [ 6 ], the notions of equivalence considered in this paper coincide. A counterexample is presented here showing that this coincidence is not always true.
{"title":"Improving E. Cartan considerations on the invariance of nonholonomic mechanics","authors":"W. M. Oliva, Gláucio Terra","doi":"10.3934/JGM.2019022","DOIUrl":"https://doi.org/10.3934/JGM.2019022","url":null,"abstract":"This paper concerns an intrinsic formulation of nonholonomic mechanics. Our point of departure is the paper [ 6 ], by Koiller et al., revisiting E. Cartan's address at the International Congress of Mathematics held in 1928 at Bologna, Italy ([ 3 ]). Two notions of equivalence for nonholonomic mechanical systems begin{document}$ ( {mathsf{{M}}}, {{mathsf{{g}}}}, {mathscr{D}}) $end{document} are introduced and studied. According to [ 6 ], the notions of equivalence considered in this paper coincide. A counterexample is presented here showing that this coincidence is not always true.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80509335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：