  We construct Parseval wavelet frames in $L^2(M)$ for a general Riemannian
manifold $M$ and we show the existence of wavelet unconditional frames in
$L^p(M)$ for $1 < p <\infty$. This is made possible thanks to smooth orthogonal
projection decomposition of the identity operator on $L^2(M)$, which was
recently proven by the authors in arXiv:1803.03634. We also show a
characterization of Triebel-Lizorkin $\mathbf F_{p,q}^s(M)$ and Besov $\mathbf
B_{p,q}^s(M)$ spaces on compact manifolds in terms of magnitudes of
coefficients of Parseval wavelet frames. We achieve this by showing that
Hestenes operators are bounded on manifolds $M$ with bounded geometry.
