  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 indexed
by 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 of
the extremes of locally stationary $\chi$-fields on manifolds.
