{"title":"D’Atri空间和半球、管和圆柱体的总标量曲率。","authors":"Balázs Csikós, Amr Elnashar, Márton Horváth","doi":"10.1007/s13163-022-00444-z","DOIUrl":null,"url":null,"abstract":"<p><p>Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about <i>geodesic</i> segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D'Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D'Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.</p>","PeriodicalId":49605,"journal":{"name":"Revista Matematica Complutense","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10471713/pdf/","citationCount":"0","resultStr":"{\"title\":\"D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders.\",\"authors\":\"Balázs Csikós, Amr Elnashar, Márton Horváth\",\"doi\":\"10.1007/s13163-022-00444-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about <i>geodesic</i> segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D'Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D'Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.</p>\",\"PeriodicalId\":49605,\"journal\":{\"name\":\"Revista Matematica Complutense\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10471713/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Revista Matematica Complutense\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s13163-022-00444-z\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2022/10/10 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Revista Matematica Complutense","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13163-022-00444-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2022/10/10 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders.
Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D'Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D'Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.
期刊介绍:
Revista Matemática Complutense is an international research journal supported by the School of Mathematics at Complutense University in Madrid. It publishes high quality research and survey articles across pure and applied mathematics. Fields of interests include: analysis, differential equations and applications, geometry, topology, algebra, statistics, computer sciences and astronomy. This broad interest is reflected in our interdisciplinary editorial board which is comprised of over 30 internationally esteemed researchers in diverse areas.
The Editorial Board of Revista Matemática Complutense organizes the “Santaló Lecture”, a yearly event where a distinguished mathematician is invited to present a lecture at Complutense University and contribute to the journal. Past lecturers include: Charles T.C. Wall, Jack K. Hale, Hans Triebel, Marcelo Viana, Narayanswamy Balakrishnan, Nigel Kalton, Alfio Quarteroni, David E. Edmunds, Giuseppe Buttazzo, Juan L. Vázquez, Eduard Feireisl, Nigel Hitchin, Lajos Horváth, Hélène Esnault, Luigi Ambrosio, Ignacio Cirac and Bernd Sturmfels. The Santaló Lecturer for 2019 will be Noel Cressie from National Institute for Applied Statistics Research Australia (NIASRA), University of Wollongong.