  Given i.i.d. observations uniformly distributed on a closed manifold
$\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the
associated empirical graph Laplacian based on a Gaussian kernel. Our main
results are non-asymptotic error bounds, showing that the eigenvalues and
eigenspaces of the empirical graph Laplacian are close to the eigenvalues and
eigenspaces of the Laplace-Beltrami operator of $\mathcal{M}$. In our analysis,
we connect the empirical graph Laplacian to kernel principal component
analysis, and consider the heat kernel of $\mathcal{M}$ as reproducing kernel
feature map. This leads to novel points of view and allows to leverage results
for empirical covariance operators in infinite dimensions.
