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Let  = {(),  ∈ } be a centered Gaussian random field, where  is a smooth Riemannian manifold. For a suitable compact subset , we obtain approximations to the excursion probabilities , as , for two cases: (i)  is smooth and isotropic; (ii)  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 . These extend the results in Cheng and Xiao (Bernoulli , 1113–1130 ) from spheres to general Riemannian manifolds.

Excursion probabilities of isotropic and locally isotropic Gaussian random fields on manifolds