Pub Date : 2024-10-01DOI: 10.1016/j.geomphys.2024.105332
Luis Hernández-Corbato , Javier Martínez-Aguinaga
We study the global topology of the space of loops of contactomorphisms of a non-orderable closed contact manifold . We filter by a quantitative measure of the “positivity” of the loops and describe the topology of in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of are subgroups of the homotopy groups of the subspace of positive loops . We obtain analogous results for the space of loops of Legendrian submanifolds in .
我们研究了非有序闭合接触流形 (M2n+1,ξ) 的接触同构环空间 L 的全局拓扑。我们用回路 "正向性 "的定量度量对 L 进行过滤，并用过滤的子空间描述 L 的拓扑。特别是，我们证明了 L 的同调群是正循环子空间 L+ 的同调群的子群。我们还得到了 (M2n+1,ξ) 中 Legendrian 子曼形环空间的类似结果。
{"title":"On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds","authors":"Luis Hernández-Corbato , Javier Martínez-Aguinaga","doi":"10.1016/j.geomphys.2024.105332","DOIUrl":"10.1016/j.geomphys.2024.105332","url":null,"abstract":"<div><div>We study the global topology of the space <span><math><mi>L</mi></math></span> of loops of contactomorphisms of a non-orderable closed contact manifold <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>ξ</mi><mo>)</mo></math></span>. We filter <span><math><mi>L</mi></math></span> by a quantitative measure of the “positivity” of the loops and describe the topology of <span><math><mi>L</mi></math></span> in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of <span><math><mi>L</mi></math></span> are subgroups of the homotopy groups of the subspace of positive loops <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>. We obtain analogous results for the space of loops of Legendrian submanifolds in <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>ξ</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422309","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 : 2024-09-27DOI: 10.1016/j.geomphys.2024.105331
Jian Qiu
The exceptional compact hermitian symmetric space EIII is the quotient . We introduce the Plücker coordinates which give an embedding of EIII into as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.
Our motivation is to understand EIII as the complex projective octonion plane , whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of .
We further decompose into -orbits: , where is an open -orbit and is the complexification of , whereas has co-dimension 1, thus EIII could be more appropriately denoted as . This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2].
{"title":"Plücker coordinates and the Rosenfeld planes","authors":"Jian Qiu","doi":"10.1016/j.geomphys.2024.105331","DOIUrl":"10.1016/j.geomphys.2024.105331","url":null,"abstract":"<div><div>The exceptional compact hermitian symmetric space EIII is the quotient <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>/</mo><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>10</mn><mo>)</mo><msub><mrow><mo>×</mo></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msub><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. We introduce the Plücker coordinates which give an embedding of EIII into <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>26</mn></mrow></msup></math></span> as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.</div><div>Our motivation is to understand EIII as the complex projective octonion plane <span><math><mo>(</mo><mi>C</mi><mo>⊗</mo><mi>O</mi><mo>)</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>.</div><div>We further decompose <span><math><mi>X</mi><mo>=</mo><mrow><mi>EIII</mi></mrow></math></span> into <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-orbits: <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>Y</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, where <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∼</mo><msub><mrow><mo>(</mo><mi>O</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>C</mi></mrow></msub></math></span> is an open <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-orbit and is the complexification of <span><math><mi>O</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, whereas <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> has co-dimension 1, thus EIII could be more appropriately denoted as <span><math><mover><mrow><msub><mrow><mo>(</mo><mi>O</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>C</mi></mrow></msub></mrow><mo>‾</mo></mover></math></span>. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer <span><span>[2]</span></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.geomphys.2024.105329
A. Alexandrov , B. Bychkov , P. Dunin-Barkowski , M. Kazarian , S. Shadrin
For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the n-point functions produced by the topological recursion on these curves via the n-point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions.
对于给定的光谱曲线，我们构建了交映对偶光谱曲线族，并证明了一个明确的公式，通过原始曲线上的 n 点函数表达了这些曲线上拓扑递归产生的 n 点函数。作为推论，我们证明了产生函数的广义完全简单映射的拓扑递归。
{"title":"Topological recursion, symplectic duality, and generalized fully simple maps","authors":"A. Alexandrov , B. Bychkov , P. Dunin-Barkowski , M. Kazarian , S. Shadrin","doi":"10.1016/j.geomphys.2024.105329","DOIUrl":"10.1016/j.geomphys.2024.105329","url":null,"abstract":"<div><div>For a given spectral curve, we construct a family of <em>symplectic dual</em> spectral curves for which we prove an explicit formula expressing the <em>n</em>-point functions produced by the topological recursion on these curves via the <em>n</em>-point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.geomphys.2024.105330
Adeeba Zaidi, Gauree Shanker
In this paper, we investigate the behavior of semi-invariant Riemannian maps taking Sasakian structure as total manifolds satisfying Ricci soliton equation, to Riemannian manifolds. We establish necessary and sufficient conditions for the cases when fibers and are Einstein. Further, we calculate scalar curvature for , fibers and total manifolds. Also, we derive some inequalities for semi-invariant Riemannian maps from Sasakian space forms satisfying Ricci soliton equation, to Riemannian manifolds. We construct some examples in support of assumed maps.
{"title":"Semi-invariant Riemannian maps from Sasakian manifolds endowed with Ricci soliton structure","authors":"Adeeba Zaidi, Gauree Shanker","doi":"10.1016/j.geomphys.2024.105330","DOIUrl":"10.1016/j.geomphys.2024.105330","url":null,"abstract":"<div><div>In this paper, we investigate the behavior of semi-invariant Riemannian maps taking Sasakian structure as total manifolds satisfying Ricci soliton equation, to Riemannian manifolds. We establish necessary and sufficient conditions for the cases when fibers and <span><math><mi>r</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><msub><mrow><mi>F</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> are Einstein. Further, we calculate scalar curvature for <span><math><mi>r</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><msub><mrow><mi>F</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>, fibers and total manifolds. Also, we derive some inequalities for semi-invariant Riemannian maps from Sasakian space forms satisfying Ricci soliton equation, to Riemannian manifolds. We construct some examples in support of assumed maps.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422310","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 : 2024-09-21DOI: 10.1016/j.geomphys.2024.105327
Tobias Fritz
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra for which the module of derivations of is finitely generated projective. Over as the base ring, this class of algebras includes the algebra of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion makes into an algebraifold with as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.
Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.
人们经常注意到，微分几何的许多基本概念，如连接的定义，在本质上都是纯代数的。在此，我们回顾并扩展了微分几何全代数公式的现有工作，这些公式不需要底层流形。我们指的是交换代数 A，A 的导数模块是有限生成的射影。以 R 为基环，这一类代数包括流形 M 上光滑函数的代数 C∞(M)，解析函数也是如此。一个重要的不同例子是 M 上广义函数的科隆博代数，它使分布微分几何学成为我们形式主义的一个实例。另一个例子是光滑微分几何的纤维化版本，因为任何光滑潜入 M→N 都会使 C∞(M)成为以 C∞(N)为基环的代数折叠。在任何特征为零的域 k 上，例子包括光滑仿射变体上的正则函数代数以及任何函数域。我们用代数折叠来发展微分几何，包括张量、连接、曲率、大地线，并简要考虑广义相对论。
{"title":"Differential geometry and general relativity with algebraifolds","authors":"Tobias Fritz","doi":"10.1016/j.geomphys.2024.105327","DOIUrl":"10.1016/j.geomphys.2024.105327","url":null,"abstract":"<div><div>It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of <em>algebraifold</em>, by which we mean a commutative algebra <span><math><mi>A</mi></math></span> for which the module of derivations of <span><math><mi>A</mi></math></span> is finitely generated projective. Over <span><math><mi>R</mi></math></span> as the base ring, this class of algebras includes the algebra <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span> of smooth functions on a manifold <em>M</em>, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on <em>M</em>, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion <span><math><mi>M</mi><mo>→</mo><mi>N</mi></math></span> makes <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span> into an algebraifold with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>N</mi><mo>)</mo></math></span> as the base ring. Over any field <em>k</em> of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.</div><div>Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319853","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 : 2024-09-20DOI: 10.1016/j.geomphys.2024.105325
Dong Shen, Jiancheng Liu
In this paper, by using the strong maximum principle, we present a necessary and sufficient conditions for constructing gradient -almost Ricci solitons with warped product structures, and give examples of particular solutions of the PDEs that arise from our construction. Also, we prove nonexistence results for gradient -almost Ricci solitons on warped product manifolds under some natural assumptions concerning the warping function or gradient vector field.
{"title":"Gradient h˜-almost Ricci solitons on warped product manifolds","authors":"Dong Shen, Jiancheng Liu","doi":"10.1016/j.geomphys.2024.105325","DOIUrl":"10.1016/j.geomphys.2024.105325","url":null,"abstract":"<div><div>In this paper, by using the strong maximum principle, we present a necessary and sufficient conditions for constructing gradient <span><math><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>-almost Ricci solitons with warped product structures, and give examples of particular solutions of the PDEs that arise from our construction. Also, we prove nonexistence results for gradient <span><math><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>-almost Ricci solitons on warped product manifolds under some natural assumptions concerning the warping function or gradient vector field.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311765","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 : 2024-09-19DOI: 10.1016/j.geomphys.2024.105326
Hacen Zelaci
In this paper, we show a strange duality isomorphism at level one for the space of generalized theta functions on the moduli spaces of alternating anti-invariant vector bundles in the ramified case. These anti-invariant vector bundles constitute one of the non-trivial examples of parahoric -torsors, where is a twisted (not generically split) parahoric group scheme.
{"title":"Strange duality at level one for alternating vector bundles","authors":"Hacen Zelaci","doi":"10.1016/j.geomphys.2024.105326","DOIUrl":"10.1016/j.geomphys.2024.105326","url":null,"abstract":"<div><div>In this paper, we show a strange duality isomorphism at level one for the space of generalized theta functions on the moduli spaces of alternating anti-invariant vector bundles in the ramified case. These anti-invariant vector bundles constitute one of the non-trivial examples of parahoric <span><math><mi>G</mi></math></span>-torsors, where <span><math><mi>G</mi></math></span> is a twisted (not generically split) parahoric group scheme.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314213","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 : 2024-09-19DOI: 10.1016/j.geomphys.2024.105328
R.S. Kraußhar , M. Ferreira , N. Vieira , M.M. Rodrigues
In the development of function theory in octonions, the non-associativity property produces an additional associator term when applying the Stokes formula. To take the non-associativity into account, particular intrinsic weight factors are implemented in the definition of octonion-valued inner products to ensure the existence of a reproducing Bergman kernel. This Bergman projection plays a pivotal role in the -space decomposition demonstrated in this paper for octonion-valued functions. In the unit ball, we explicitly show that the intrinsic weight factor is crucial to obtain the reproduction property and that the latter precisely compensates an additional associator term that otherwise appears when leaving out the weight factor.
Furthermore, we study an octonionic Teodorescu transform and show how it is related to the unweighted version of the Bergman transform and establish some operator relations between these transformations. We apply two different versions of the Borel-Pompeiu formulae that naturally arise in the context of the non-associativity. Next, we use the octonionic Teodorescu transform to establish a suitable octonionic generalization of the Ahlfors-Beurling operator, also known as the Π-operator. We prove an integral representation formula that presents a unified representation for the Π-operator arising in all prominent hypercomplex function theories. Then we describe some basic mapping properties arising in context with the -space decomposition discussed before.
Finally, we explore several applications of the octonionic Π-operator. Initially, we demonstrate its utility in solving the octonionic Beltrami equation, which characterizes generalized quasi-conformal maps from to in a specific analytical sense. Subsequently, analogous results are presented for the hyperbolic octonionic Dirac operator acting on the right half-space of . Lastly, we discuss how the octonionic Teodorescu transform and the Bergman projection can be employed to solve an eight-dimensional Stokes problem in the non-associative octonionic setting.
{"title":"The Teodorescu and the Π-operator in octonionic analysis and some applications","authors":"R.S. Kraußhar , M. Ferreira , N. Vieira , M.M. Rodrigues","doi":"10.1016/j.geomphys.2024.105328","DOIUrl":"10.1016/j.geomphys.2024.105328","url":null,"abstract":"<div><div>In the development of function theory in octonions, the non-associativity property produces an additional associator term when applying the Stokes formula. To take the non-associativity into account, particular intrinsic weight factors are implemented in the definition of octonion-valued inner products to ensure the existence of a reproducing Bergman kernel. This Bergman projection plays a pivotal role in the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-space decomposition demonstrated in this paper for octonion-valued functions. In the unit ball, we explicitly show that the intrinsic weight factor is crucial to obtain the reproduction property and that the latter precisely compensates an additional associator term that otherwise appears when leaving out the weight factor.</div><div>Furthermore, we study an octonionic Teodorescu transform and show how it is related to the unweighted version of the Bergman transform and establish some operator relations between these transformations. We apply two different versions of the Borel-Pompeiu formulae that naturally arise in the context of the non-associativity. Next, we use the octonionic Teodorescu transform to establish a suitable octonionic generalization of the Ahlfors-Beurling operator, also known as the Π-operator. We prove an integral representation formula that presents a unified representation for the Π-operator arising in all prominent hypercomplex function theories. Then we describe some basic mapping properties arising in context with the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-space decomposition discussed before.</div><div>Finally, we explore several applications of the octonionic Π-operator. Initially, we demonstrate its utility in solving the octonionic Beltrami equation, which characterizes generalized quasi-conformal maps from <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>8</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>8</mn></mrow></msup></math></span> in a specific analytical sense. Subsequently, analogous results are presented for the hyperbolic octonionic Dirac operator acting on the right half-space of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>8</mn></mrow></msup></math></span>. Lastly, we discuss how the octonionic Teodorescu transform and the Bergman projection can be employed to solve an eight-dimensional Stokes problem in the non-associative octonionic setting.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142327238","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 : 2024-09-17DOI: 10.1016/j.geomphys.2024.105324
Valentin Lychagin
In this paper, we study the thermodynamics of homogeneous mixtures in equilibrium. From the perspective of thermodynamics, substances are understood as Legendre submanifolds, which are equipped with a Riemannian structure in addition. We refer to these as Legendre-Riemannian manifolds. This Legendre structure reflects the law of conservation of energy, while the Riemannian structure corresponds to the second central moment of measurement of extensive quantities, indicating that we only consider stable states. Thermodynamic processes, such as chemical reactions, correspond to contact vector fields that preserve the law of energy conservation, or are contact. The presence of a Riemannian structure distinguishes between three classes of processes: positive, which increase the metric; neutral, which preserve the metric; and negative, which decrease the metric. We provide a detailed description of the processes and suggest a method for finding state equations for a homogeneous mixture in mechanical or chemical equilibrium.
{"title":"On thermodynamic processes, state equations and critical phenomena for homogeneous mixtures","authors":"Valentin Lychagin","doi":"10.1016/j.geomphys.2024.105324","DOIUrl":"10.1016/j.geomphys.2024.105324","url":null,"abstract":"<div><div>In this paper, we study the thermodynamics of homogeneous mixtures in equilibrium. From the perspective of thermodynamics, substances are understood as Legendre submanifolds, which are equipped with a Riemannian structure in addition. We refer to these as Legendre-Riemannian manifolds. This Legendre structure reflects the law of conservation of energy, while the Riemannian structure corresponds to the second central moment of measurement of extensive quantities, indicating that we only consider stable states. Thermodynamic processes, such as chemical reactions, correspond to contact vector fields that preserve the law of energy conservation, or are contact. The presence of a Riemannian structure distinguishes between three classes of processes: positive, which increase the metric; neutral, which preserve the metric; and negative, which decrease the metric. We provide a detailed description of the processes and suggest a method for finding state equations for a homogeneous mixture in mechanical or chemical equilibrium.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318763","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 : 2024-09-17DOI: 10.1016/j.geomphys.2024.105322
Julia Bernatska
In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated.
The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic -function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where , as well as other ℘-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera 2 and 3.
{"title":"Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces","authors":"Julia Bernatska","doi":"10.1016/j.geomphys.2024.105322","DOIUrl":"10.1016/j.geomphys.2024.105322","url":null,"abstract":"<div><div>In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated.</div><div>The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic <span><math><msub><mrow><mo>℘</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span>-function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where <span><math><msub><mrow><mo>℘</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span>, as well as other ℘-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera 2 and 3.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319852","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}
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