We utilize generalized moving least squares (GMLS) to develop meshfree
techniques for discretizing hydrodynamic flow problems on manifolds. We use
exterior calculus to formulate incompressible hydrodynamic equations in the
Stokesian regime and handle the divergence-free constraints via a generalized
vector potential. This provides less coordinate-centric descriptions and
enables the development of efficient numerical methods and splitting schemes
for the fourth-order governing equations in terms of a system of second-order
elliptic operators. Using a Hodge decomposition, we develop methods for
manifolds having spherical topology. We show the methods exhibit high-order
convergence rates for solving hydrodynamic flows on curved surfaces. The
methods also provide general high-order approximations for the metric,
curvature, and other geometric quantities of the manifold and associated
exterior calculus operators. The approaches also can be utilized to develop
high-order solvers for other scalar-valued and vector-valued problems on
manifolds.