{"title":"克莱沃公式的推广及其应用","authors":"Vadym Koval","doi":"arxiv-2409.02895","DOIUrl":null,"url":null,"abstract":"The main purpose of this article is to study conditions for a curve on a\nsubmanifold $M\\subset\\mathbb{R}^n$, constructed in a particular way involving\nthe Euclidean distance to $M$, to be a geodesic. We also present the naturally\narising generalization of Clairaut's formula needed for the generalization of\nthe main result to higher dimensions.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":"70 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A generalization of Clairaut's formula and its applications\",\"authors\":\"Vadym Koval\",\"doi\":\"arxiv-2409.02895\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The main purpose of this article is to study conditions for a curve on a\\nsubmanifold $M\\\\subset\\\\mathbb{R}^n$, constructed in a particular way involving\\nthe Euclidean distance to $M$, to be a geodesic. We also present the naturally\\narising generalization of Clairaut's formula needed for the generalization of\\nthe main result to higher dimensions.\",\"PeriodicalId\":501113,\"journal\":{\"name\":\"arXiv - MATH - Differential Geometry\",\"volume\":\"70 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Differential Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.02895\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Differential Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02895","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A generalization of Clairaut's formula and its applications
The main purpose of this article is to study conditions for a curve on a
submanifold $M\subset\mathbb{R}^n$, constructed in a particular way involving
the Euclidean distance to $M$, to be a geodesic. We also present the naturally
arising generalization of Clairaut's formula needed for the generalization of
the main result to higher dimensions.