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Depending on a parameter $h\in (0,1]$, let $\{X_h(\mathbf{t})$,$\mathbf{t}\in\mathcal{M}_h\}$ be a class of centered Gaussian fields indexedby compact manifolds $\mathcal{M}_h$. For locally stationary Gaussian fields$X_h$, we study the asymptotic excursion probabilities of $X_h$ on$\mathcal{M}_h$. Two cases are considered: (i) $h$ is fixed and (ii)$h\rightarrow0$. These results are extended to obtain the limit behaviors ofthe extremes of locally stationary $\chi$-fields on manifolds.

Extremes of locally stationary Gaussian and chi fields on manifolds