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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.14356 |
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| _version_ | 1866909295975596032 |
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| author | Su, Zhe Tong, Yiying Wei, Guo-Wei |
| author_facet | Su, Zhe Tong, Yiying Wei, Guo-Wei |
| contents | The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. 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 introduce 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 $L^2$-orthogonality and the accurate cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14356 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topology-preserving Hodge Decomposition in the Eulerian Representation Su, Zhe Tong, Yiying Wei, Guo-Wei Differential Geometry The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. 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 introduce 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 $L^2$-orthogonality and the accurate cohomology. |
| title | Topology-preserving Hodge Decomposition in the Eulerian Representation |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2408.14356 |