  Recent advances in cell biology and experimental techniques using
reconstituted cell extracts have generated significant interest in
understanding how geometry and topology influence active fluid dynamics. In
this work, we present a comprehensive continuous theory and computational
method to explore the dynamics of active nematic fluids on arbitrary surfaces
without topological constraints. The fluid velocity and nematic order parameter
are represented as the sections of the complex line bundle of a 2-manifold. We
introduce the Levi-Civita connection and surface curvature form within the
framework of complex line bundles. By adopting this geometric approach, we
introduce a gauge-invariant discretization method that preserves the continuous
local-to-global theorems in differential geometry. We establish a nematic
Laplacian on complex functions that can accommodate fractional topological
charges through the covariant derivative on the complex nematic representation.
We formulate advection of the nematic field based on a unifying definition of
the Lie derivative, resulting in a stable geometric semi-Lagrangian
discretization scheme for transport by the flow. In general, the proposed
surface-based method offers an efficient and stable means to investigate the
influence of local curvature and global topology on the 2D hydrodynamics of
active nematic systems. Moreover, the complex line representation of the
nematic field and the unifying Lie advection present a systematic approach for
generalizing our method to active $k$-atic systems.
