If one has a a compact Riemannian surface $\Sigma$ (ie. a Riemannian manifold of dimension $2$), the Ricci curvature can be written as
$R_{ij} = \frac{1}{2} R g_{ij}$.
In other words, the curvature can be expressed by one single function. The scalar curvature is equal to double the Gaussian curvature $K$, so one can also re-write
$R_{ij} = K g_{ij}$.
Does this mean that the curvature of a smooth embedded close curve in $\Sigma$ is also determined by the Gaussian curvature $K$ for the surface by Gauss-Bonnet theorem?