I am reading Do Carmo's Differential geometry of curves and surfaces currently. I'm on chapter's 4-4 and 4-5 where the 4-4 talks about parallel transport/covariant derivatives and the 4-5 talks about the gauss bonnet theorem.
Before proving the local gauss-bonnet theorem, the author introduces the theorem of turning tangents which is said to be a result in topology. The theorem being
$\sum_{i=0}^k (\phi_i (t+1) - \phi_i(t)) + \sum_{i=0}^k \theta_i = \pm2\pi$
I think I intuitively understand this formula. The first summation really just measures the change in the angle of the tangent vector to the curve as it moves along one of it's piecewise parts. The second term for theta is the external angle that accounts for the sharp turns at these piecewise junctions.
What I don't understand is the definition of phi. Phi is chosen to be the angle between the tangent vector the curve and one of the coordinate basis vectors, for example, $x_u$.
Why would this be the case? I would think that the correct definition for phi is that it would measure the angle between the tangent vector to the curve, and any parallel direction being transported along the curve. Mainly because when I'm thinking about this picture in flat space, we measure the turning angle as the angle between the tangent vector and any parallel direction.