  We are concerned with decentralized optimization over a compact submanifold,
where the loss functions of local datasets are defined by their respective
local datasets. A key challenge in decentralized optimization is mitigating the
communication bottleneck, which primarily involves two strategies: achieving
consensus and applying communication compression. Existing
projection/retraction-type algorithms rely on multi-step consensus to attain
both consensus and optimality. Due to the nonconvex nature of the manifold
constraint, it remains an open question whether the requirement for multi-step
consensus can be reduced to single-step consensus. We address this question by
carefully elaborating on the smoothness structure and the asymptotic
1-Lipschitz continuity associated with the manifold constraint. Furthermore, we
integrate these insights with a communication compression strategy to propose a
communication-efficient gradient algorithm for decentralized manifold
optimization problems, significantly reducing per-iteration communication
costs. Additionally, we establish an iteration complexity of
$\mathcal{O}(\epsilon^{-1})$ to find an $\epsilon$-stationary point, which
matches the complexity in the Euclidean setting. Numerical experiments
demonstrate the efficiency of the proposed method in comparison to
state-of-the-art approaches.
