  We study the convergence of the graph Laplacian of a random geometric graph
generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$
as the sample size $n$ increases and the neighborhood size $h$ tends to zero.
We show that eigenvalues and eigenvectors of the graph Laplacian converge with
a rate of $O\Big(\big(\frac{\log n}{n}\big)^\frac{1}{2m}\Big)$ to the
eigenvalues and eigenfunctions of the weighted Laplace-Beltrami operator of
$M$.
  No information on the submanifold $M$ is needed in the construction of the
graph or the "out-of-sample extension" of the eigenvectors. Of independent
interest is a generalization of the rate of convergence of empirical measures
on submanifolds in $R^d$ in infinity transportation distance.
