  This paper provides a para-differential calculus toolbox on compact Lie
groups and homogeneous spaces. It helps to understand non-local, nonlinear
partial differential operators with low regularity on manifolds with high
symmetry. In particular, the paper provides a para-linearization formula for
the Dirichlet-Neumann operator of a distorted 2-sphere, a key ingredient in
understanding long-time behaviour of spherical capillary water waves. As an
initial application, the paper provides a new proof of local well-posedness for
spherical capillary water waves equation under weaker regularity assumptions
compared to previous results.
