Recently ı have started to read the book of I. Chavel " Riemannian Geometry". I have a question that keep my mind busy. In R^n, We can derive classical Green's and divergence theorem from the Stokes theorem. Even we do have Green's formula 1-2. (We newer say Stoke's Theorem to them. )
Suppose D is a bounded domain with smooth boundary in n-dimensional Riemannian manifold.
Should we call Green's formulas as "Stokes Theorem" if we are studying in a Riemannian manifold? Most of the book says Green's formula in Differential Geometry. Why are not they calling "by Stokes theorem" while they are applying integration by part? Is there any specific reason? or is it too obvious?
Many thanks for your answer.