Journal of Geometry and Physics最新文献

{"title":"On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds","authors":"Luis Hernández-Corbato ,&nbsp;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":"206 ","pages":"Article 105332"},"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}

{"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":"206 ","pages":"Article 105331"},"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}

{"title":"Topological recursion, symplectic duality, and generalized fully simple maps","authors":"A. Alexandrov ,&nbsp;B. Bychkov ,&nbsp;P. Dunin-Barkowski ,&nbsp;M. Kazarian ,&nbsp;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":"206 ","pages":"Article 105329"},"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}

{"title":"Semi-invariant Riemannian maps from Sasakian manifolds endowed with Ricci soliton structure","authors":"Adeeba Zaidi,&nbsp;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":"206 ","pages":"Article 105330"},"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}

{"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":"206 ","pages":"Article 105327"},"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}

{"title":"Gradient h˜-almost Ricci solitons on warped product manifolds","authors":"Dong Shen,&nbsp;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":"207 ","pages":"Article 105325"},"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}

{"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":"207 ","pages":"Article 105326"},"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}

{"title":"The Teodorescu and the Π-operator in octonionic analysis and some applications","authors":"R.S. Kraußhar ,&nbsp;M. Ferreira ,&nbsp;N. Vieira ,&nbsp;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":"206 ","pages":"Article 105328"},"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}

{"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":"207 ","pages":"Article 105324"},"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}

{"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":"206 ","pages":"Article 105322"},"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}