  We prove the Quantitative Fatou Theorem for Lipschitz domains on complete
Riemannian manifolds. This requires extending the $\varepsilon$-approximation
lemma to the manifold setting. Our studies apply to harmonic functions, as well
as to a class of A-harmonic functions on manifolds. The presented results
extend works by Dahlberg, Garnett, Bortz and Hofmann.
