Topology-preserving Hodge decomposition in the Eulerian representation

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 $L^2$ -orthogonality and the accurate cohomology. The proposed method paves the way for manifold topological analysis (MTA) and manifold topological learning (MTL).