Pub Date : 2025-09-24DOI: 10.1016/j.geomphys.2025.105654
Esteban Andruchow , Gabriel Larotonda , Lázaro Recht
Let be a component of the Grassmann manifold of a -algebra, presented as the unitary orbit of a given orthogonal projection . There are several natural connections on this manifold, and we first show that they all agree (in the presence of a finite trace in , when we give the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of for the spectral rectifiable distance, and also the conjugate tangent locus of along a geodesic. Furthermore, for each tangent vector V at P, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all.
{"title":"Conjugate points in the Grassmann manifold of a C⁎-algebra","authors":"Esteban Andruchow , Gabriel Larotonda , Lázaro Recht","doi":"10.1016/j.geomphys.2025.105654","DOIUrl":"10.1016/j.geomphys.2025.105654","url":null,"abstract":"<div><div>Let <figure><img></figure> be a component of the Grassmann manifold of a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra, presented as the unitary orbit of a given orthogonal projection <figure><img></figure>. There are several natural connections on this manifold, and we first show that they all agree (in the presence of a finite trace in <span><math><mi>A</mi></math></span>, when we give <figure><img></figure> the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of <figure><img></figure> for the spectral rectifiable distance, and also the conjugate tangent locus of <figure><img></figure> along a geodesic. Furthermore, for each tangent vector <em>V</em> at <em>P</em>, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105654"},"PeriodicalIF":1.2,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-23DOI: 10.1016/j.geomphys.2025.105653
Andrei Căldăraru , Tony Pantev , Eric Sharpe , Benjamin Sung , Xingyang Yu
In this paper we explore noninvertible symmetries in general (not necessarily rational) SCFTs and their topological B-twists for Calabi-Yau manifolds. We begin with a detailed overview of defects in the topological B model. For trivial reasons, all defects in the topological B model are topological operators, and define (often noninvertible) symmetries of that topological field theory, but only a subset remain topological in the physical (i.e., untwisted) theory. For a generic target space Calabi-Yau X, we discuss geometric realizations of those defects, as simultaneously A- and B-twistable complex Lagrangian and complex coisotropic branes on , and discuss their fusion products. To be clear, the possible noninvertible symmetries in the B model are more general than can be described with fusion categories. That said, we do describe realizations of some Tambara-Yamagami categories in the B model for an elliptic curve target, and also argue that elliptic curves can not admit Fibonacci or Haagerup structures. We also discuss how decomposition is realized in this language.
{"title":"Noninvertible symmetries in the B model TFT","authors":"Andrei Căldăraru , Tony Pantev , Eric Sharpe , Benjamin Sung , Xingyang Yu","doi":"10.1016/j.geomphys.2025.105653","DOIUrl":"10.1016/j.geomphys.2025.105653","url":null,"abstract":"<div><div>In this paper we explore noninvertible symmetries in general (not necessarily rational) SCFTs and their topological B-twists for Calabi-Yau manifolds. We begin with a detailed overview of defects in the topological B model. For trivial reasons, all defects in the topological B model are topological operators, and define (often noninvertible) symmetries of that topological field theory, but only a subset remain topological in the physical (i.e., untwisted) theory. For a generic target space Calabi-Yau <em>X</em>, we discuss geometric realizations of those defects, as simultaneously A- and B-twistable complex Lagrangian and complex coisotropic branes on <span><math><mi>X</mi><mo>×</mo><mi>X</mi></math></span>, and discuss their fusion products. To be clear, the possible noninvertible symmetries in the B model are more general than can be described with fusion categories. That said, we do describe realizations of some Tambara-Yamagami categories in the B model for an elliptic curve target, and also argue that elliptic curves can not admit Fibonacci or Haagerup structures. We also discuss how decomposition is realized in this language.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105653"},"PeriodicalIF":1.2,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-22DOI: 10.1016/j.geomphys.2025.105650
M. Huzaifa Yaseen , Rida Hashmi , Najla A. Mohammed , Hala A Hejazi
The Lie symmetry method offers a systematic approach for analyzing and solving differential equations by identifying continuous transformations that preserve their structure. In this study, we investigate a general system of two nonlinear second-order elliptic partial differential equations using Lie symmetry techniques. We compute the equivalence transformations for the system, which serve as the foundation for deriving differential invariants. Specifically, we establish both joint differential invariants that are obtained under transformations of dependent and independent variables along with semi-differential invariants, derived solely from transformations of dependent variables. These invariants play a crucial role in reducing the system to its simplest possible form while retaining its essential features. By applying these differential invariants, we present reduced forms of various nonlinear systems of elliptic partial differential equations, demonstrating the effectiveness of the method in simplifying complex equations. Our results highlight the utility of Lie symmetry analysis in deriving invariant structures and facilitating the systematic reduction of coupled nonlinear systems of partial differential equations.
{"title":"Differential invariants of systems of two nonlinear elliptic partial differential equations by Lie symmetry method","authors":"M. Huzaifa Yaseen , Rida Hashmi , Najla A. Mohammed , Hala A Hejazi","doi":"10.1016/j.geomphys.2025.105650","DOIUrl":"10.1016/j.geomphys.2025.105650","url":null,"abstract":"<div><div>The Lie symmetry method offers a systematic approach for analyzing and solving differential equations by identifying continuous transformations that preserve their structure. In this study, we investigate a general system of two nonlinear second-order elliptic partial differential equations using Lie symmetry techniques. We compute the equivalence transformations for the system, which serve as the foundation for deriving differential invariants. Specifically, we establish both joint differential invariants that are obtained under transformations of dependent and independent variables along with semi-differential invariants, derived solely from transformations of dependent variables. These invariants play a crucial role in reducing the system to its simplest possible form while retaining its essential features. By applying these differential invariants, we present reduced forms of various nonlinear systems of elliptic partial differential equations, demonstrating the effectiveness of the method in simplifying complex equations. Our results highlight the utility of Lie symmetry analysis in deriving invariant structures and facilitating the systematic reduction of coupled nonlinear systems of partial differential equations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105650"},"PeriodicalIF":1.2,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-18DOI: 10.1016/j.geomphys.2025.105649
Xinru Cao , Zafar Normatov , Bakhrom Omirov , Jie Ruan
This paper investigates nilpotent and solvable structures in generalized Poisson algebras, establishing analogues of Engel's and Lie's theorems within this context. We present several constructions of generalized Poisson algebras, including those derived from null-filiform and filiform associative commutative algebras, and explore extensions through unit adjunction and generalized Wronskian Lie algebras. Using polarization techniques, we establish fundamental equivalences between algebraic structures and characterize admissible algebras. Finally, we provide a complete classification of complex nilpotent generalized Poisson algebras up to dimension three.
{"title":"On generalized Poisson algebras: Solvability and constructions","authors":"Xinru Cao , Zafar Normatov , Bakhrom Omirov , Jie Ruan","doi":"10.1016/j.geomphys.2025.105649","DOIUrl":"10.1016/j.geomphys.2025.105649","url":null,"abstract":"<div><div>This paper investigates nilpotent and solvable structures in generalized Poisson algebras, establishing analogues of Engel's and Lie's theorems within this context. We present several constructions of generalized Poisson algebras, including those derived from null-filiform and filiform associative commutative algebras, and explore extensions through unit adjunction and generalized Wronskian Lie algebras. Using polarization techniques, we establish fundamental equivalences between algebraic structures and characterize admissible algebras. Finally, we provide a complete classification of complex nilpotent generalized Poisson algebras up to dimension three.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105649"},"PeriodicalIF":1.2,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145160293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-18DOI: 10.1016/j.geomphys.2025.105651
Bahar Doğan Yazıcı
In this study, we investigate the geometry of generalized rectifying ruled surfaces in the 3-dimensional Lie group . We construct geometric structures such as singular point sets, cylindrical surfaces, striction curves, developable surfaces, geodesic and asymptotic curves, as well as the Gauss and mean curvatures of generalized rectifying ruled surfaces in . Then, we present the shape operator matrix and some related characterizations of developable generalized rectifying ruled surfaces in the 3-dimensional Lie group . We also discuss how generalized rectifying ruled surfaces in 3-dimensional Lie groups correspond, in special cases, to tangent developable ruled surfaces, binormal ruled surfaces, and rectifying ruled surfaces both in 3-dimensional Lie groups and in 3-dimensional Euclidean space.
{"title":"On the construction of generalized rectifying ruled surfaces in 3-dimensional Lie groups","authors":"Bahar Doğan Yazıcı","doi":"10.1016/j.geomphys.2025.105651","DOIUrl":"10.1016/j.geomphys.2025.105651","url":null,"abstract":"<div><div>In this study, we investigate the geometry of generalized rectifying ruled surfaces in the 3-dimensional Lie group <span><math><mi>G</mi></math></span>. We construct geometric structures such as singular point sets, cylindrical surfaces, striction curves, developable surfaces, geodesic and asymptotic curves, as well as the Gauss and mean curvatures of generalized rectifying ruled surfaces in <span><math><mi>G</mi></math></span>. Then, we present the shape operator matrix and some related characterizations of developable generalized rectifying ruled surfaces in the 3-dimensional Lie group <span><math><mi>G</mi></math></span>. We also discuss how generalized rectifying ruled surfaces in 3-dimensional Lie groups correspond, in special cases, to tangent developable ruled surfaces, binormal ruled surfaces, and rectifying ruled surfaces both in 3-dimensional Lie groups and in 3-dimensional Euclidean space.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"217 ","pages":"Article 105651"},"PeriodicalIF":1.2,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145120940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-16DOI: 10.1016/j.geomphys.2025.105648
Shujing Pan, Yong Wei
In this paper, we consider the anisotropic α-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power , if the initial hypersurface is complete noncompact and locally uniformly convex, then there exists a complete, noncompact, smooth and strictly convex solution of the flow which is defined for all positive time.
{"title":"Anisotropic Gauss curvature flow of complete non-compact graphs","authors":"Shujing Pan, Yong Wei","doi":"10.1016/j.geomphys.2025.105648","DOIUrl":"10.1016/j.geomphys.2025.105648","url":null,"abstract":"<div><div>In this paper, we consider the anisotropic <em>α</em>-Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, if the initial hypersurface is complete noncompact and locally uniformly convex, then there exists a complete, noncompact, smooth and strictly convex solution of the flow which is defined for all positive time.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105648"},"PeriodicalIF":1.2,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-11DOI: 10.1016/j.geomphys.2025.105645
Daniele Corradetti , David Chester , Raymond Aschheim , Klee Irwin
In this paper we present a general setting for aperiodic Jordan algebras arising from icosahedral quasicrystals that are obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In these hypotheses, we show the existence of an aperiodic Jordan algebra structure whose generators are in one-to-one correspondence with elements of the quasicrystal. Moreover, if the acceptance window enjoys a non-crystallographic symmetry arising from or then the resulting Jordan algebra enjoys the same or symmetry. Finally, we present as special cases some examples of Jordan algebras over a Fibonacci-chain quasicrystal, a Penrose tiling, and the Elser-Sloane quasicrystal.
{"title":"Jordan algebras over icosahedral cut-and-project quasicrystals","authors":"Daniele Corradetti , David Chester , Raymond Aschheim , Klee Irwin","doi":"10.1016/j.geomphys.2025.105645","DOIUrl":"10.1016/j.geomphys.2025.105645","url":null,"abstract":"<div><div>In this paper we present a general setting for aperiodic Jordan algebras arising from icosahedral quasicrystals that are obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In these hypotheses, we show the existence of an aperiodic Jordan algebra structure whose generators are in one-to-one correspondence with elements of the quasicrystal. Moreover, if the acceptance window enjoys a non-crystallographic symmetry arising from <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> then the resulting Jordan algebra enjoys the same <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> symmetry. Finally, we present as special cases some examples of Jordan algebras over a Fibonacci-chain quasicrystal, a Penrose tiling, and the Elser-Sloane quasicrystal.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"217 ","pages":"Article 105645"},"PeriodicalIF":1.2,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-11DOI: 10.1016/j.geomphys.2025.105646
Valentin Lychagin
In this paper, we apply the method of geometrization of random vectors [1] to turbulent media, which we understand as random vector fields on base manifolds. This gives rise to various geometric structures on the tangent as well as cotangent bundles. Among these, the most important is the Mahalanobis metric on the tangent bundle, which allows us to obtain all the necessary ingredients for implementing the scheme [2] to the description of flows in turbulent media. As an illustration, we consider the applications to flows of real gases based on Maxwell–Boltzmann statistics.
{"title":"On geometry of turbulent flows","authors":"Valentin Lychagin","doi":"10.1016/j.geomphys.2025.105646","DOIUrl":"10.1016/j.geomphys.2025.105646","url":null,"abstract":"<div><div>In this paper, we apply the method of geometrization of random vectors <span><span>[1]</span></span> to turbulent media, which we understand as random vector fields on base manifolds. This gives rise to various geometric structures on the tangent as well as cotangent bundles. Among these, the most important is the Mahalanobis metric on the tangent bundle, which allows us to obtain all the necessary ingredients for implementing the scheme <span><span>[2]</span></span> to the description of flows in turbulent media. As an illustration, we consider the applications to flows of real gases based on Maxwell–Boltzmann statistics.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"217 ","pages":"Article 105646"},"PeriodicalIF":1.2,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-11DOI: 10.1016/j.geomphys.2025.105644
Sharief Deshmukh , Nasser Bin Turki , Hemangi Madhusudan Shah , Gabriel-Eduard Vîlcu
<div><div>Ricci solitons are stationary solutions of a famous PDE for Riemannian metrics, known under the name of Ricci flow equation. An almost Ricci soliton is a remarkable generalization of Ricci solitons by allowing the soliton constant in Ricci flow equation to be a smooth function. In the present paper, we focuss our study on the most important class of almost Ricci solitons, namely gradient Ricci almost solitons <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> with potential function <em>σ</em> and associated function <em>f</em>, abbreviated as <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>. On a nontrivial <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>, these two functions <em>σ</em> and <em>f</em> together with scalar curvature <em>τ</em> play a significant role. Among the basic properties of a connected <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>, it has been observed that there exists a smooth function <em>δ</em> called the connector of the <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> as it connects the gradients of the potential function <em>σ</em> and the associated function <em>f</em>, respectively. In our first result it is shown that a nontrivial <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> with connector <em>δ</em> gives a generalized soliton, thus establishing an unexpected duality. In our second result, we show that a compact and connected nontrivial <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> with connector <span><math><mi>δ</mi><mo>=</mo><mo>−</mo><mi>c</mi></math></span>, for a positive constant <em>c</em>, and a suitable lower bound on the integral of the Ricci curvature <span><math><mi>R</mi><mi>i</mi><mi>c</mi><mrow><mo>(</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>)</mo></mrow></math></span> is isometric to the <em>n</em>-sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo></math></span> and the converse too is shown to hold. In the third result it is established that a
{"title":"Some basic properties of Ricci almost solitons","authors":"Sharief Deshmukh , Nasser Bin Turki , Hemangi Madhusudan Shah , Gabriel-Eduard Vîlcu","doi":"10.1016/j.geomphys.2025.105644","DOIUrl":"10.1016/j.geomphys.2025.105644","url":null,"abstract":"<div><div>Ricci solitons are stationary solutions of a famous PDE for Riemannian metrics, known under the name of Ricci flow equation. An almost Ricci soliton is a remarkable generalization of Ricci solitons by allowing the soliton constant in Ricci flow equation to be a smooth function. In the present paper, we focuss our study on the most important class of almost Ricci solitons, namely gradient Ricci almost solitons <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> with potential function <em>σ</em> and associated function <em>f</em>, abbreviated as <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>. On a nontrivial <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>, these two functions <em>σ</em> and <em>f</em> together with scalar curvature <em>τ</em> play a significant role. Among the basic properties of a connected <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span>, it has been observed that there exists a smooth function <em>δ</em> called the connector of the <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> as it connects the gradients of the potential function <em>σ</em> and the associated function <em>f</em>, respectively. In our first result it is shown that a nontrivial <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> with connector <em>δ</em> gives a generalized soliton, thus establishing an unexpected duality. In our second result, we show that a compact and connected nontrivial <em>GRRAS</em> <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>g</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> with connector <span><math><mi>δ</mi><mo>=</mo><mo>−</mo><mi>c</mi></math></span>, for a positive constant <em>c</em>, and a suitable lower bound on the integral of the Ricci curvature <span><math><mi>R</mi><mi>i</mi><mi>c</mi><mrow><mo>(</mo><mi>∇</mi><mi>σ</mi><mo>,</mo><mi>∇</mi><mi>σ</mi><mo>)</mo></mrow></math></span> is isometric to the <em>n</em>-sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo></math></span> and the converse too is shown to hold. In the third result it is established that a ","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"217 ","pages":"Article 105644"},"PeriodicalIF":1.2,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-11DOI: 10.1016/j.geomphys.2025.105647
Miao-Miao Xie, Shou-Fu Tian, Xing-Jie Yan
In this work, by the squared eigenfunctions of the spectral problem for the modified Hunter-Saxton equation, we derive the generalized Fourier transform and the symplectic basis for the equation. First, we present the symmetry and the asymptotic behavior of the Jost solutions and the scattering data from the inverse scattering transform. Then the completeness relations of the Jost solutions and the squared eigenfunctions are derived by constructing two meromorphic functions, from which we can derive the generalized Fourier transform. Finally, we verified that a set of variables defined by the scattering data and the squared eigenfunctions form the symplectic basis of the phase space, which gives the description in symplectic geometry for the modified Hunter-Saxton equation.
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