To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I am confused about the relation between these two systems.
While the first system is consistent with the curvature of a curve, does the curvature tensor has anything to do with the curvature which means something like the curvature of a curve? What's more, does the torsion tensor have anything to do with the torsion which means something like the torsion of a curve?
The principle curvatures are extrinsic quantities, the depend on the embedding of the manifold into some exterior manifold, and describe how the embedded manifold curves in that space.
The curvature tensor of a Riemannian (or Lorentzian) manifold is an intrinsic quantity which measures to which extend covariant derivatives commute.