From Classical to Discrete: Exploring Curvature Computation for Topological Preservation and High-Dimensional Extensions

Jinyang Cai

2025

Abstract

The Gauss-Bonnet theorem in differential geometry connects global topological invariants with local curvature, with classical formulations by Allendoerfer Weil and Chern influencing theoretical physics and mathematics. Recent discrete differential geometry advances compute curvature via vertex summation on triangulated surfaces, suitable for computational use. This paper clarifies classical foundations and evaluates computational efficiency/accuracy in discrete curvature quantification. Using a four-stage method (dataset prep, curvature computation, evaluation metrics, hybrid validation), it applies Meyer’s discrete exterior calculus (DEC) and Thurston's angle defect model, extending to 3D tetrahedral meshes. Results show discrete methods offer 2–3 orders faster computation but have RMSE-varying geometric accuracy with mesh resolution, while classical integration ensures topological consistency (TFI=0). The 3D extension confirms topological fidelity on regular grids. The study highlights DEC’s efficiency-accuracy balance and discrete methods’ non-smooth region challenges, bridging computational geometry and network science via a hybrid framework. Limitations include underdeveloped high-dimensional theory, hybrid method overhead, and singularity-induced errors. Future work should address theoretical generalization, deep learning-integrated algorithms, and quantum geometry/topological machine learning applications to enhance the theorem’s computational utility and theoretical understanding.

Paper Citation

in Harvard Style

Cai J. (2025). From Classical to Discrete: Exploring Curvature Computation for Topological Preservation and High-Dimensional Extensions. In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA; ISBN 978-989-758-774-0, SciTePress, pages 303-310. DOI: 10.5220/0013824600004708

in Bibtex Style

@conference{iampa25,
author={Jinyang Cai},
title={From Classical to Discrete: Exploring Curvature Computation for Topological Preservation and High-Dimensional Extensions},
booktitle={Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA},
year={2025},
pages={303-310},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013824600004708},
isbn={978-989-758-774-0},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA
TI - From Classical to Discrete: Exploring Curvature Computation for Topological Preservation and High-Dimensional Extensions
SN - 978-989-758-774-0
AU - Cai J.
PY - 2025
SP - 303
EP - 310
DO - 10.5220/0013824600004708
PB - SciTePress