Surface Theory Revisited
If M is a surface in R3, it is easy to see that M 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 M to R3, and then projecting the result orthogonally back onto the tangent spaces of M. 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 M induced by the inner product on R3, hence it is the Levi-Civita connection of this metric.
Read more about this topic: Affine Connection
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