Pub Date : 2026-01-08DOI: 10.1016/j.geomphys.2026.105758
Jie Fei , Jun Wang
In this paper, we investigate the classification problem for holomorphic two-spheres of constant curvature in the complex Grassmann manifold under additional geometric conditions. By considering the (1,0)-part of μ-th covariant differential about the second fundamental form denoted by , , its norm denoted by , we establish the following results: for unramified holomorphic two-spheres with constant curvature and constant squared norm of the second fundamental form, the quantity is necessarily constant. Moreover, under additional conditions that is positive and is identically zero, we obtain a complete classification of such holomorphic two-spheres.
{"title":"On classification of holomorphic two-spheres of constant curvature in the complex Grassmann manifold G(3,6)","authors":"Jie Fei , Jun Wang","doi":"10.1016/j.geomphys.2026.105758","DOIUrl":"10.1016/j.geomphys.2026.105758","url":null,"abstract":"<div><div>In this paper, we investigate the classification problem for holomorphic two-spheres of constant curvature in the complex Grassmann manifold <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></span> under additional geometric conditions. By considering the (1,0)-part of <em>μ</em>-th covariant differential about the second fundamental form denoted by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mi>μ</mi></mrow></msub></math></span>, <span><math><mi>μ</mi><mo>≥</mo><mn>1</mn></math></span>, its norm denoted by <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mi>μ</mi></mrow></msub><mo>|</mo></math></span>, we establish the following results: for unramified holomorphic two-spheres with constant curvature and constant squared norm of the second fundamental form, the quantity <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>|</mo></math></span> is necessarily constant. Moreover, under additional conditions that <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>|</mo></math></span> is positive and <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>|</mo></math></span> is identically zero, we obtain a complete classification of such holomorphic two-spheres.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105758"},"PeriodicalIF":1.2,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980338","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 : 2026-01-07DOI: 10.1016/j.geomphys.2025.105750
Paul Norbury
In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces . This allows us to prove via algebraic geometry a recursion between the volumes of moduli spaces of super hyperbolic surfaces previously proven via super geometry techniques by Stanford and Witten. The recursion between the volumes of moduli spaces of super hyperbolic surfaces is proven to be equivalent to the property that a generating function for the intersection numbers of a natural collection of cohomology classes with tautological classes on is a KdV tau function. This is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem, which relates a generating function for the intersection numbers of tautological classes on to KdV, using volumes of moduli spaces of hyperbolic surfaces.
{"title":"Enumerative geometry via the moduli space of super Riemann surfaces","authors":"Paul Norbury","doi":"10.1016/j.geomphys.2025.105750","DOIUrl":"10.1016/j.geomphys.2025.105750","url":null,"abstract":"<div><div>In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. This allows us to prove via algebraic geometry a recursion between the volumes of moduli spaces of super hyperbolic surfaces previously proven via super geometry techniques by Stanford and Witten. The recursion between the volumes of moduli spaces of super hyperbolic surfaces is proven to be equivalent to the property that a generating function for the intersection numbers of a natural collection of cohomology classes <span><math><msub><mrow><mi>Θ</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with tautological classes on <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> is a KdV tau function. This is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem, which relates a generating function for the intersection numbers of tautological classes on <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> to KdV, using volumes of moduli spaces of hyperbolic surfaces.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105750"},"PeriodicalIF":1.2,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038569","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 : 2025-12-31DOI: 10.1016/j.geomphys.2025.105748
Dongha Lee
We extend the framework of submanifolds in Riemannian geometry to Riemann-Cartan geometry, which addresses connections with torsion. This procedure naturally introduces a 2-form on submanifolds associated with the nontrivial ambient torsion, whose Hodge dual plays the role of an imaginary counterpart to mean curvature for surfaces in a Riemann-Cartan 3-manifold. We observe that this complex-valued geometric quantity interacts with a number of other geometric concepts including the Hopf differential and the Gauss map, which generalizes classical minimal surface theory.
{"title":"Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry","authors":"Dongha Lee","doi":"10.1016/j.geomphys.2025.105748","DOIUrl":"10.1016/j.geomphys.2025.105748","url":null,"abstract":"<div><div>We extend the framework of submanifolds in Riemannian geometry to Riemann-Cartan geometry, which addresses connections with torsion. This procedure naturally introduces a 2-form on submanifolds associated with the nontrivial ambient torsion, whose Hodge dual plays the role of an imaginary counterpart to mean curvature for surfaces in a Riemann-Cartan 3-manifold. We observe that this complex-valued geometric quantity interacts with a number of other geometric concepts including the Hopf differential and the Gauss map, which generalizes classical minimal surface theory.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105748"},"PeriodicalIF":1.2,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940685","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 : 2025-12-31DOI: 10.1016/j.geomphys.2025.105747
Keita Mori, Yasuhiro Wakabayashi
This note studies -opers arising from generalized hypergeometric differential equations in prime characteristic p. We prove that these opers are rigid within the class of dormant opers. By combining this rigidity result with previous work in the enumerative geometry of dormant opers, we obtain a complete and explicit description of the 2d TQFTs that compute the number of dormant -opers for primes .
{"title":"Generalized hypergeometric equations and 2d TQFT for dormant opers in characteristic ≤7","authors":"Keita Mori, Yasuhiro Wakabayashi","doi":"10.1016/j.geomphys.2025.105747","DOIUrl":"10.1016/j.geomphys.2025.105747","url":null,"abstract":"<div><div>This note studies <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-opers arising from generalized hypergeometric differential equations in prime characteristic <em>p</em>. We prove that these opers are rigid within the class of dormant opers. By combining this rigidity result with previous work in the enumerative geometry of dormant opers, we obtain a complete and explicit description of the 2d TQFTs that compute the number of dormant <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-opers for primes <span><math><mi>p</mi><mo>≤</mo><mn>7</mn></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105747"},"PeriodicalIF":1.2,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940687","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 : 2025-12-30DOI: 10.1016/j.geomphys.2025.105743
Grigorios Giotopoulos , Hisham Sati , Urs Schreiber
We give a modernized and streamlined review, aimed at mathematical physicists, of the origin and nature of the super Lie-algebra known as the (“hidden”) M-algebra, which arises somewhat subtly in analysis of 11D supergravity. Following arguments that this (hidden) M-algebra serves in fact as the maximal super-exceptional tangent space for 11D supergravity, we particularly make explicit here its integration to a (super-Lie) group. This is equipped with a left-invariant extension of the “decomposed” M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries.
As a simple but consequential application, we highlight how to describe lattice subgroups of the hidden M-group that allow to toroidially compactify also the “hidden” dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality.
In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the “decomposed” M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice.
Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations which we touch on along the way is the dimensional reduction of the hidden M-algebra to a “hidden IIA-algebra” which in [45] we have explained as the exceptional extension of the T-duality doubled super-spacetime.
{"title":"The hidden M-group","authors":"Grigorios Giotopoulos , Hisham Sati , Urs Schreiber","doi":"10.1016/j.geomphys.2025.105743","DOIUrl":"10.1016/j.geomphys.2025.105743","url":null,"abstract":"<div><div>We give a modernized and streamlined review, aimed at mathematical physicists, of the origin and nature of the super Lie-algebra known as the (“hidden”) <em>M-algebra</em>, which arises somewhat subtly in analysis of 11D supergravity. Following arguments that this (hidden) M-algebra serves in fact as the maximal super-exceptional tangent space for 11D supergravity, we particularly make explicit here its integration to a (super-Lie) <em>group</em>. This is equipped with a left-invariant extension of the “decomposed” M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries.</div><div>As a simple but consequential application, we highlight how to describe lattice subgroups <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mo>≤</mo><mn>528</mn></mrow></msup></math></span> of the hidden M-group that allow to toroidially compactify also the “hidden” dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality.</div><div>In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the “decomposed” M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice.</div><div>Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations which we touch on along the way is the dimensional reduction of the hidden M-algebra to a “hidden IIA-algebra” which in <span><span>[45]</span></span> we have explained as the exceptional extension of the T-duality doubled super-spacetime.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105743"},"PeriodicalIF":1.2,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898029","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 : 2025-12-30DOI: 10.1016/j.geomphys.2025.105746
Guolin An , Guangxiang Su
Let be a closed Riemannian manifold of dimension n, and let F be an integrable subbundle of TM. Let be the leafwise scalar curvature associated to . Let E be an oriented flat vector bundle. We show that if either TM or F is spin, and TM carries a metric satisfying that , the leafwise scalar curvature along F, is positive everywhere, then , where is the Hirzebruch -class of TM and is the Euler class of E. This extends the generalization of the Lichnerowicz vanishing theorem concerning the Euler class proved by Yu and Zhang to the case of foliations.
设（M,gTM）为n维的封闭黎曼流形，设F为TM的可积子束。设kF为与gF=gTM|F相关的叶向标量曲率。设E是一个有方向的平面向量束。我们证明了如果TM或F是自旋，并且TM携带一个度量gTM，满足沿F的叶向标量曲率kF处处为正，则< a - (TM)e(e),[M] > =0，其中a - (TM)是TM的Hirzebruch a -类，e(e)是e的欧拉类。这将Yu和Zhang证明的关于欧拉类的Lichnerowicz消失定理推广到叶分的情况。
{"title":"Positive scalar curvature on foliations and the Euler class","authors":"Guolin An , Guangxiang Su","doi":"10.1016/j.geomphys.2025.105746","DOIUrl":"10.1016/j.geomphys.2025.105746","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>M</mi><mo>,</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>T</mi><mi>M</mi></mrow></msup><mo>)</mo></math></span> be a closed Riemannian manifold of dimension <em>n</em>, and let <em>F</em> be an integrable subbundle of <em>TM</em>. Let <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>F</mi></mrow></msup></math></span> be the leafwise scalar curvature associated to <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>F</mi></mrow></msup><mo>=</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>T</mi><mi>M</mi></mrow></msup><msub><mrow><mo>|</mo></mrow><mrow><mi>F</mi></mrow></msub></math></span>. Let <em>E</em> be an oriented flat vector bundle. We show that if either <em>TM</em> or <em>F</em> is spin, and <em>TM</em> carries a metric <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>T</mi><mi>M</mi></mrow></msup></math></span> satisfying that <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>F</mi></mrow></msup></math></span>, the leafwise scalar curvature along <em>F</em>, is positive everywhere, then <span><math><mo>〈</mo><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>T</mi><mi>M</mi><mo>)</mo><mi>e</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>,</mo><mo>[</mo><mi>M</mi><mo>]</mo><mo>〉</mo><mo>=</mo><mn>0</mn></math></span>, where <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>T</mi><mi>M</mi><mo>)</mo></math></span> is the Hirzebruch <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-class of <em>TM</em> and <span><math><mi>e</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> is the Euler class of <em>E</em>. This extends the generalization of the Lichnerowicz vanishing theorem concerning the Euler class proved by Yu and Zhang to the case of foliations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105746"},"PeriodicalIF":1.2,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898028","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 : 2025-12-30DOI: 10.1016/j.geomphys.2025.105749
A.V. Smilga
It was recently found that a necessary and sufficient condition for a Kähler manifold to be hyperkähler reads(1) where is a complex metric, is a symplectic matrix and C is a positive constant.
In this note, we give a simple explicit proof of this fact.
{"title":"On the multidimensional heavenly equation","authors":"A.V. Smilga","doi":"10.1016/j.geomphys.2025.105749","DOIUrl":"10.1016/j.geomphys.2025.105749","url":null,"abstract":"<div><div>It was recently found that a necessary and sufficient condition for a Kähler manifold to be hyperkähler reads<span><span><span>(1)</span><span><math><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><msub><mrow><mi>h</mi></mrow><mrow><mi>j</mi><mover><mrow><mi>l</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><msup><mrow><mi>Ω</mi></mrow><mrow><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover><mover><mrow><mi>l</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msup><mspace></mspace><mo>=</mo><mspace></mspace><mi>C</mi><msub><mrow><mi>Ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo></mrow></math></span></span></span> where <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> is a complex metric, <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span> is a symplectic matrix and <em>C</em> is a positive constant.</div><div>In this note, we give a simple explicit proof of this fact.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105749"},"PeriodicalIF":1.2,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940683","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 : 2025-12-29DOI: 10.1016/j.geomphys.2025.105745
Mark D. Hamilton , Yael Karshon , Takahiko Yoshida
We give a simple proof that, for a pre-quantized compact symplectic manifold with a Lagrangian torus fibration, its Riemann–Roch number coincides with its number of Bohr–Sommerfeld fibres. This can be viewed as an instance of the “independence of polarization” phenomenon of geometric quantization. The base space for such a fibration acquires a so-called integral-integral affine structure. The proof uses the following simple fact, whose proof is trickier than we expected: on a compact integral-integral affine manifold, the total volume is equal to the number of integer points.
{"title":"Integral-integral affine geometry, geometric quantization, and Riemann–Roch","authors":"Mark D. Hamilton , Yael Karshon , Takahiko Yoshida","doi":"10.1016/j.geomphys.2025.105745","DOIUrl":"10.1016/j.geomphys.2025.105745","url":null,"abstract":"<div><div>We give a simple proof that, for a pre-quantized compact symplectic manifold with a Lagrangian torus fibration, its Riemann–Roch number coincides with its number of Bohr–Sommerfeld fibres. This can be viewed as an instance of the “independence of polarization” phenomenon of geometric quantization. The base space for such a fibration acquires a so-called integral-integral affine structure. The proof uses the following simple fact, whose proof is trickier than we expected: on a compact integral-integral affine manifold, the total volume is equal to the number of integer points.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105745"},"PeriodicalIF":1.2,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940686","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 : 2025-12-29DOI: 10.1016/j.geomphys.2025.105742
Rutwig Campoamor-Stursberg , Oscar Carballal , Francisco J. Herranz
We propose an adaptation of the notion of scaling symmetries for the case of Lie–Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie–Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra . It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley–Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie–Hamilton systems through a curvature-dependent reduction.
{"title":"Contact Lie systems on Riemannian and Lorentzian spaces: From scaling symmetries to curvature-dependent reductions","authors":"Rutwig Campoamor-Stursberg , Oscar Carballal , Francisco J. Herranz","doi":"10.1016/j.geomphys.2025.105742","DOIUrl":"10.1016/j.geomphys.2025.105742","url":null,"abstract":"<div><div>We propose an adaptation of the notion of scaling symmetries for the case of Lie–Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie–Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra <span><math><mi>s</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>R</mi><mo>)</mo></math></span>. It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley–Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie–Hamilton systems through a curvature-dependent reduction.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105742"},"PeriodicalIF":1.2,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884910","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 : 2025-12-29DOI: 10.1016/j.geomphys.2025.105744
Kiyoon Eum
In this paper, we study the full asymptotic expansion of the partition functions of determinantal point processes defined on a polarized Kähler manifold. We show that the coefficients of the expansion are given by geometric functionals on Kähler metrics satisfying the cocycle identity, whose first variations can be expressed through the TYZ expansion coefficients of the Bergman kernel. In particular, these functionals naturally generalize the Mabuchi functional in Kähler geometry and the Liouville functional on Riemann surfaces. We further show that Futaki-type holomorphic invariants obstruct the existence of critical points of these geometric functionals, extending Lu's formula. We also verify that certain formulas remain valid up to the third coefficient without assuming polarization. Finally, we discuss the relation of our results to the quantum Hall effect (QHE), where the determinantal point process provides a microscopic model. In particular, we recover the higher-dimensional effective Chern-Simons actions derived in the physics literature and confirm a conjecture of Klevtsov on the form of the partition function asymptotics.
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