{"title":"准共形变形产生的平面场的弗罗贝尼斯可整性定理","authors":"Slobodan N. Simić","doi":"10.1016/j.difgeo.2024.102202","DOIUrl":null,"url":null,"abstract":"<div><div>We generalize the classical Frobenius integrability theorem to plane fields of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span>, a regularity class introduced by Reimann <span><span>[9]</span></span> for vector fields in Euclidean spaces. Reimann showed that a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span> vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span> plane field (defined in a suitable way) in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is integrable, with integral manifolds of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>Q</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"97 ","pages":"Article 102202"},"PeriodicalIF":0.6000,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Frobenius integrability theorem for plane fields generated by quasiconformal deformations\",\"authors\":\"Slobodan N. Simić\",\"doi\":\"10.1016/j.difgeo.2024.102202\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We generalize the classical Frobenius integrability theorem to plane fields of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span>, a regularity class introduced by Reimann <span><span>[9]</span></span> for vector fields in Euclidean spaces. Reimann showed that a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span> vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>Q</mi></mrow></msup></math></span> plane field (defined in a suitable way) in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is integrable, with integral manifolds of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>Q</mi></mrow></msup></math></span>.</div></div>\",\"PeriodicalId\":51010,\"journal\":{\"name\":\"Differential Geometry and its Applications\",\"volume\":\"97 \",\"pages\":\"Article 102202\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-11-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0926224524000950\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224524000950","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
A Frobenius integrability theorem for plane fields generated by quasiconformal deformations
We generalize the classical Frobenius integrability theorem to plane fields of class , a regularity class introduced by Reimann [9] for vector fields in Euclidean spaces. Reimann showed that a vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive plane field (defined in a suitable way) in is integrable, with integral manifolds of class .
期刊介绍：
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.
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