Journal of the
Korean Mathematical Society JKMS

By using the method of equivalence of E. Cartan we calculate the local scalar invariants for Riemannian 2-maniolds. We define also a notion of local invariants for submanifolds in ${\Bbb R}^{n+d},$ $n \ge 2$, $d \ge 1$, in terms of the symmetry of the local isometric embedding equations of Riemannian $n$-manifolds into ${\Bbb R}^{n+d}$. We show that the local invariants obtained by the Cartan's method are the intrinsic expressions of the local invariants in our sense in the cases of surfaces in ${\Bbb R}^3$.

Keywords: equivalence problem, prolongation, local invariants

MSC numbers: 53A55, 58A20