Numerical calculation method for function integration on submanifolds of $\mathbb{R}^n$ or compact Riemannian manifolds

In this paper, we present a method for digitally representing the "volume element" and calculating the integral of a function on compact hypersurfaces with or without boundary, and low-dimensional submanifolds in $\mathbb{R}^n$. We also extend such calculation to hypersurfaces in compact Riemannnian manifolds.