  Let $X= \{X(p), p\in M\}$ be a centered Gaussian random field, where $M$ is a
smooth Riemannian manifold. For a suitable compact subset $D\subset M$, we
obtain the approximations to excursion probability $\mathbb{P}\{\sup_{p\in D}
X(p) \ge u \}$, as $u\to \infty$, for two cases: (i) $X$ is smooth and
isotropic; (ii) $X$ is non-smooth and locally isotropic. For case (i), the
expected Euler characteristic approximation is formulated explicitly; while for
case (ii), it is shown that the asymptotics is similar to Pickands'
approximation on Euclidean space which involves Pickands' constant and the
volume of $D$. These extend the results in \citep{Cheng:2014} from sphere to
general Riemannian manifolds.
