I am reading the chapter "Absolute Differentiation and Displacement" in "Differential Geometry" by Erwin Kresyzig. I have trouble understanding theorem 76.1 which says that
The displacement of Levi-Civita, i.e. $$\frac{da^{\alpha}}{ds} + a^{\beta} \Gamma_{\beta \sigma}^{\alpha}\dot{u}^{\sigma} =0 \quad \quad \text{(1)}$$, is independent of the choice of the curve, iff the surface under consideration is a developable surface.
All the notations are tensorial; $u^1,u^2$ are the coordinates on the surface S which is in a 3-d Euclidean space; $a^1(s),a^2(s) $ are the components of vectors $a(s)$ that makes a constant angle with the curve C which is parameterized by $s$.
Proof:
Step-1: $\frac{da^{\alpha}}{ds} + a^{\beta} \Gamma_{\beta \sigma}^{\alpha}\dot{u}^{\sigma} =0 $ is independent of the choice of curve, then the covariant derivative must be zero i.e. $$a^{\alpha}_{,\sigma} = 0$$
Step-2: The integrability condition for $a^{\alpha}_{,\sigma} = 0$ is given by $$a^{\gamma}R^{\alpha}_{\gamma\tau\sigma} = 0$$, which is obtained from Ricci's identity $a^{\alpha}_{,\sigma\mu} - a^{\alpha}_{,\mu\sigma} = -a^{\tau}R^{\alpha}_{\tau\sigma\mu}$
Question 1: Actually, the author derived the equation (1) for a vector $a^{\alpha}$ that makes a constant angle with a Geodesic on a surface S. I am not sure if the same equation holds if the curve is not a geodesic - i.e. if a vector $a^{\alpha}$ makes a constant angle with a curve C, is equation (1) still valid?
Question 2: Why should the covariant derivative be zero in Step-1?
Question 3: Can you please explain the integrability condition in Step -2?
Thanks in advance!
P.S.: I am an Engineer and this is just my first reading of differential geometry, so I am not familiar with a lot of usual terminologies.
Edit: I think I got the solution to Question 1. $$\frac{da^{\alpha}}{ds} = \frac{\partial a^{\alpha}} { \partial u^{\sigma}} \dot{u}^{\sigma}. $$ Equation (1) becomes $$ a^{\alpha}_{,\sigma} \dot{u}^{\sigma} = 0 $$ This means that the covariant derivative $a^{\alpha}_{,\sigma}$ is normal to the surface at all points or the covariant derivative is $a^{\alpha}_{,\sigma} = 0$