Geometry of the Covariant Derivative

Question

Taking the standard covariant derivative from tensor calculus $ \nabla_{\mu}T_{s} =T_{s|\mu}=T_{s,\mu} $. Could this be geometrically interpreted as the directional derivate of a tensor that has extra terms to compensate for the fact that the manifold in question may not be flat, whereas in the flat case it just generalizes to the regular partial derivate of multivariable calculus?

Answer 1

Yes. If the manifold is embedded in Euclidean space and has the induced metric, then the covariant derivative is exactly the projection of the directional derivative in ambient space onto the tangent hyperplane.