Covariant derivative from connections

[lavinia]

On a 2 dimensional Riemannian manifold how does one derive the covariant derivative from the connection 1 form on the tangent unit circle bundle?

 

[Ben Niehoff]

Its the same formula in any dimension. Let $X = X^a e_a$ be a vector field, where $e_a$ is an orthonormal frame. Then

$$\nabla X = (D X^a) \otimes e_a = (d X^a + \omega^a{}_b X^b) \otimes e_a$$