{"title":"Topology-preserving Hodge decomposition in the Eulerian representation.","authors":"Zhe Su, Yiying Tong, Guo-Wei Wei","doi":"10.4310/bpam.250908175047","DOIUrl":"10.4310/bpam.250908175047","url":null,"abstract":"<p><p>The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. It has significant applications in data science. Despite extensive research in the past few decades, topology-preserving Hodge decomposition of scalar and vector fields on manifolds with boundaries in the Eulerian representation remains a challenge due to the implicit incorporation of appropriate topology-preserving boundary conditions. In this work, we present a comprehensive 5-component topology-preserving Hodge decomposition that unifies normal and tangential components in the Cartesian representation. Implicit representations of planar and volumetric regions defined by level-set functions have been developed. Numerical experiments on various objects, including single-cell RNA velocity, validate the effectiveness of our approach, confirming the expected rigorous <math> <msup><mrow><mi>L</mi></mrow> <mrow><mn>2</mn></mrow> </msup> </math> -orthogonality and the accurate cohomology. The proposed method paves the way for manifold topological analysis (MTA) and manifold topological learning (MTL).</p>","PeriodicalId":521292,"journal":{"name":"Beijing journal of pure & applied mathematics","volume":"2 2","pages":"619-657"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12967212/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147380511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}