  In this work, we introduced a class of nonlocal models to accurately
approximate the Poisson model on manifolds that are embedded in high
dimensional Euclid spaces with Dirichlet boundary. In comparison to the
existing nonlocal Poisson models, instead of utilizing volumetric boundary
constraint to reduce the truncation error to its local counterpart, we rely on
the Poisson equation itself along the boundary to explicitly express the second
order normal derivative by some geometry-based terms, so that to create a new
model with $\mathcal{O}(\delta)$ truncation error along the $2\delta-$boundary
layer and $\mathcal{O}(\delta^2)$ at interior, with $\delta$ be the nonlocal
interaction horizon. Our concentration is on the construction and the
truncation error analysis of such nonlocal model. The control on the truncation
error is currently optimal among all nonlocal models, and is sufficient to
attain second order localization rate that will be derived in our subsequent
work.
