Topology-preserving Hodge decomposition in the Eulerian representation.

Area of Science:

• Differential Geometry
• Algebraic Topology
• Data Science

Background:

• Hodge decomposition is crucial for differential forms on Riemannian manifolds.
• Topology-preserving Hodge decomposition on manifolds with boundaries is challenging.
• Existing methods struggle with implicit boundary conditions.

Purpose of the Study:

• To present a comprehensive 5-component topology-preserving Hodge decomposition.
• To unify normal and tangential components in Cartesian representation.
• To address challenges in manifold topological analysis and learning.

Main Methods:

• Developed implicit representations of planar and volumetric regions using level-set functions.
• Implemented a 5-component Hodge decomposition unifying normal and tangential components.
• Utilized Cartesian representation for unified component handling.

Main Results:

• Validated the approach through numerical experiments on various objects, including single-cell RNA velocity.
• Confirmed rigorous L2-orthogonality and accurate cohomology.
• Demonstrated the effectiveness of the topology-preserving boundary conditions.

Conclusions:

• The proposed method offers a robust solution for topology-preserving Hodge decomposition.
• This work advances manifold topological analysis (MTA) and manifold topological learning (MTL).
• The approach is effective for scalar and vector fields on manifolds with boundaries.