{"title":"封闭曲面的不变分布和输运扭曲空间","authors":"Jan Bohr, Thibault Lefeuvre, Gabriel P. Paternain","doi":"10.1112/jlms.12894","DOIUrl":null,"url":null,"abstract":"<p>We study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the <i>transport twistor space</i> of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a <i>unital algebra</i>, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on <span></span><math>\n <semantics>\n <msup>\n <mi>S</mi>\n <mn>2</mn>\n </msup>\n <annotation>$\\mathbb {S}^2$</annotation>\n </semantics></math> are determined by their zeroth and first Fourier modes.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12894","citationCount":"0","resultStr":"{\"title\":\"Invariant distributions and the transport twistor space of closed surfaces\",\"authors\":\"Jan Bohr, Thibault Lefeuvre, Gabriel P. Paternain\",\"doi\":\"10.1112/jlms.12894\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the <i>transport twistor space</i> of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a <i>unital algebra</i>, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on <span></span><math>\\n <semantics>\\n <msup>\\n <mi>S</mi>\\n <mn>2</mn>\\n </msup>\\n <annotation>$\\\\mathbb {S}^2$</annotation>\\n </semantics></math> are determined by their zeroth and first Fourier modes.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12894\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12894\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12894","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们研究了闭合定向黎曼曲面单位切线束上的输运方程及其与曲面输运扭转空间(一个天然适合于大地向量场的复曲面)的联系。我们证明了在测地流下不变的纤维全形分布(在曲面上的张量层析成像中起着重要作用)形成了一个单原子代数,也就是说,这种分布的乘法是定义明确且连续的。我们还展示了纤维全纯不变分布与扭子空间上真正的全纯函数之间的自然双射对应关系,扭子空间的边界上存在多项式吹胀。此外,当曲面是阿诺索夫曲面时,我们对捻子空间上光滑到边界的全形线束进行了分类。作为我们分析的副产品,我们得到了弗拉米尼奥(Flaminio)结果的定量版本[C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735-738],该结果断言 S 2 $\mathbb {S}^2$ 上正曲度量的大地流的不变分布由它们的第零模和第一傅里叶模决定。
Invariant distributions and the transport twistor space of closed surfaces
We study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on are determined by their zeroth and first Fourier modes.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.