  We consider the problem of decentralized nonconvex optimization over a
compact submanifold, where each local agent's objective function defined by the
local dataset is smooth. Leveraging the powerful tool of proximal smoothness,
we establish local linear convergence of the projected gradient descent method
with a unit step size for solving the consensus problem over the nonconvex
compact submanifold. This serves as the basis for designing and analyzing
decentralized algorithms on manifolds. Subsequently, we propose two
decentralized methods: the decentralized projected Riemannian gradient descent
(DPRGD) and the decentralized projected Riemannian gradient tracking (DPRGT).
We establish their convergence rates of $\mathcal{O}(1/\sqrt{K})$ and
$\mathcal{O}(1/K)$, respectively, to reach a stationary point. To the best of
our knowledge, DPRGT is the first decentralized algorithm to achieve exact
convergence for solving decentralized optimization over a compact submanifold.
Beyond the linear convergence results on the consensus, two key tools developed
in the proof are the Lipschitz-type inequality of the projection operator and
the Riemannian quadratic upper bound for smooth functions on the compact
submanifold, which could be of independent interest. Finally, we demonstrate
the effectiveness of our proposed methods compared to state-of-the-art ones
through numerical experiments on eigenvalue problems and low-rank matrix
completion.
