This thesis presents a method for constructing globally consistent direction fields on discrete surfaces by defining a "trivial connection" through a sparse linear system.

This framework's latent novelty lies in its potential application to state synchronization and feature alignment problems on complex networks beyond geometry processing, particularly within the realm of distributed systems or machine learning on graphs.

Modern graph neural networks (GNNs) are powerful but often struggle with explicitly enforcing global consistency or structured alignment critical in tasks like multi-agent coordination or consistent feature propagation on complex graphs.

Integrating this discrete connection framework could provide a geometrically-principled mechanism: using the linear solve to generate a globally consistent alignment structure (the "trivial connection") that guides GNN message passing or acts as a regularization loss, especially for state spaces that form abelian groups (like phases or vectors).