High Excursion Probabilities for Gaussian Fields on Smooth Manifolds

Gaussian random fields on finite-dimensional smooth manifolds, whose variance functions reach their maximum values at smooth submanifolds, are considered, and the exact asymptotic behavior of large excursion probabilities is established. It is shown that our conditions on the behavior of the covariation and variance are best possible in the context of the classical Pickands double sum method. Applications of our asymptotic formulas to large deviations of Gaussian vector processes are considered, and some examples are given. This paper continues the previous study of the author with Kobelkov, Rodionov, and Hashorva [J. Math. Sci., 262 (2022), pp. 504--513] which was concerned with Gaussian processes and fields on manifolds with a single point of maximum of the variance.