  We consider the consensus problem in a decentralized network, focusing on a
compact submanifold that acts as a nonconvex constraint set. By leveraging the
proximal smoothness of the compact submanifold, which encompasses the local
singleton property and the local Lipschitz continuity of the projection
operator on the manifold, and establishing the connection between the
projection operator and general retraction, we show that the Riemannian
gradient descent with a unit step size has locally linear convergence if the
network has a satisfactory level of connectivity. Moreover, based on the
geometry of the compact submanifold, we prove that a convexity-like regularity
condition, referred to as the restricted secant inequality, always holds in an
explicitly characterized neighborhood around the solution set of the nonconvex
consensus problem. By leveraging this restricted secant inequality and imposing
a weaker connectivity requirement on the decentralized network, we present a
comprehensive analysis of the linear convergence of the Riemannian gradient
descent, taking into consideration appropriate initialization and step size.
Furthermore, if the network is well connected, we demonstrate that the local
Lipschitz continuity endowed by proximal smoothness is a sufficient condition
for the restricted secant inequality, thus contributing to the local error
bound. We believe that our established results will find more application in
the consensus problems over a more general proximally smooth set. Numerical
experiments are conducted to validate our theoretical findings.
