if is a surface in , it is easy to see that has a natural affine connection. from the linear connection point of view, the covariant derivative of a vector field is defined by differentiating the vector field, viewed as a map from to , and then projecting the result orthogonally back onto the tangent spaces of . it is easy to see that this affine connection is torsion-free. furthermore, it is a metric connection with respect to the riemannian metric on induced by the inner product on , hence it is the levi-civita connection of this metric.