  Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let
$\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f
\in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$
denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is
well-localized near the diagonal, in the sense that it satisfies estimates akin
to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on
$\RR^n$. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a
manner that $K_t(x,y)$ satisfies this definition, because of its localization
near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to
continuous wavelets on $\RR^n$ in $\mathcal{S}(\RR^n)$. In particular, we are
able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by
the size of their continuous ${\mathcal{S}}-$wavelet transforms, for
H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus
$\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat''
situation), we obtain two explicit approximate formulas for $K_t$, one to be
used when $t$ is large, and one to be used when $t$ is small.
