  Let $Y_i,i\geq1$, be i.i.d. random variables having values in an
$m$-dimensional manifold $\mathcal {M}\subset \mathbb{R}^d$ and consider sums
$\sum_{i=1}^n\xi(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n)$, where $\xi$ is a real
valued function defined on pairs $(y,\mathcal {Y})$, with $y\in \mathbb{R}^d$
and $\mathcal {Y}\subset \mathbb{R}^d$ locally finite. Subject to $\xi$
satisfying a weak spatial dependence and continuity condition, we show that
such sums satisfy weak laws of large numbers, variance asymptotics and central
limit theorems. We show that the limit behavior is controlled by the value of
$\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes
tangent to $\mathcal {M}$. We apply the general results to establish the limit
theory of dimension and volume content estimators, R\'{e}nyi and Shannon
entropy estimators and clique counts in the Vietoris-Rips complex on
$\{Y_i\}_{i=1}^n$.
