We know that a discrete set (for example $S=\{a,b\}$) is a $0$-manifold. But how we can define a Riemannian metric on this set. Is this trivial and don't make sense ?
Can we define the normal derivative and Laplace-Beltrami on $S$ in rigourous way ? (I guess it's just the usual derivatives taken at points).
May be the answer is obvious, but it's not clear for me.
It's all trivial.
On a zero-dimensional manifold, the tangent spaces are zero-dimensional. The only inner product on the trivial space is the trivial one $\langle 0,0\rangle = 0$ - so that's the metric.
Similarly, anything that involves derivatives will be trivial, since it maps into those trivial tangent spaces along the way. The Laplacian is the divergence of the gradient, and that gradient is just zero.
The only definition of a normal derivative I found involves embedding our manifold in a space of degree $1$ greater. In this case, that would be a line - so, yes, ordinary derivatives at those points, of a function defined on that line.