  We use a new method to give conditions for the existence of a local isometric
immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given
$n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar
fields on the manifold, satisfying a certain non-linear equation involving the
Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the
fundamental theorem of hypersurfaces. In the case of manifolds of positive
sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and
Codazzi equations to the cancelation of a set of obstructions involving the
logarithm of the Riemann curvature operator. The resulting theorem has a
structural similarity to the Weyl-Schouten theorem, suggesting a parallelism
between conformally flat $n$-manifolds and those that admit an isometric
immersion in $\mathbb{R}^{n+1}$.
