{"title":"局部保角积流形上的适应度量","authors":"Andrei Moroianu, Mihaela Pilca","doi":"10.1090/proc/16706","DOIUrl":null,"url":null,"abstract":"<p>We show that the Gauduchon metric <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g 0\"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a compact locally conformally product manifold <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M comma c comma upper D right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M,c,D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension greater than <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is adapted, in the sense that the Lee form of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g 0\"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishes on the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-flat distribution of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Adapted metrics on locally conformally product manifolds\",\"authors\":\"Andrei Moroianu, Mihaela Pilca\",\"doi\":\"10.1090/proc/16706\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that the Gauduchon metric <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"g 0\\\"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\\\"application/x-tex\\\">g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a compact locally conformally product manifold <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper M comma c comma upper D right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(M,c,D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension greater than <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is adapted, in the sense that the Lee form of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D\\\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"g 0\\\"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\\\"application/x-tex\\\">g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishes on the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D\\\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-flat distribution of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M\\\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16706\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16706","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明，维度大于 2 2 的紧凑局部保角积流形 ( M , c , D ) (M,c,D) 的高都松度量 g 0 g_0 是自适应的，即 D D 关于 g 0 g_0 的李形式在 M M 的 D D 平面分布上消失。我们还将适配度量描述为定义在共形类上的自然函数的临界点。
Adapted metrics on locally conformally product manifolds
We show that the Gauduchon metric g0g_0 of a compact locally conformally product manifold (M,c,D)(M,c,D) of dimension greater than 22 is adapted, in the sense that the Lee form of DD with respect to g0g_0 vanishes on the DD-flat distribution of MM. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.
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