Why the covariant derivative does not depend of the parametrization?

Question

I'm studying Differential Geometry using the book "Diferential Geometry of Curves and Surfaces - Manfredo P. do Carmo", and he defines covariant derivative as:

Let $S \subset \mathbb{R}^3$ a surface, $p \in S$ and $\omega:S \rightarrow \bigcup\limits_{p \in S} T_p S$ a field, such that

$$\omega(p) \in T_pS, \hspace{0.1cm} \mbox{for all $p$ $\in S$}. $$

Consider a parametrization $\sigma : U\subset \mathbb{R}^2 \rightarrow V \cap S$, with $\sigma(q) = p$, and a curve $\alpha(t) := \sigma(u(t),v(t))$, satisfying $$\alpha(0) = p $$ $$\alpha'(0) = u'(0) \frac{\partial\sigma}{\partial u}(q) +v'(0) \frac{\partial\sigma}{\partial u} (q) = y \in T_pS. $$

Using the symbols above, we can write the $\omega$ field as follows

$$\omega(\sigma(u,v)) = a(u,v) \frac{\partial\sigma}{\partial u} (u,v) + b(u,v)\frac{\partial\sigma}{\partial v}(u,v), $$ where $b, a: U \rightarrow \mathbb{R}$ $\in \mathcal{C}^{\infty} (U,\mathbb{R}) $

So, we define the covariant derivative of $\omega$ in the diretion $y$ ($D_y \omega (p)$) as:

$$D_y \omega (p) = \pi_{T_pS}\circ \left(\left.\frac{d \omega(\alpha(t))}{dt}\right\rvert_{t=0} \right)$$ $=\left((a\circ\sigma^{-1} \circ \alpha)'(0) + \Gamma_{11}^1(q) a(q) u'(0) + \Gamma_{12}^1 (q) a(q) v'(0) + \Gamma_{12}^{1}(q) b(q) u'(0) + \Gamma_{22}^1(q) b(q) v'(0) \right)\sigma_u(q) + + \left((b\circ\sigma^{-1} \circ \alpha)'(0) + \Gamma_{11}^2(q) a(q) u'(0) + \Gamma_{12}^2(q) a(q) v'(0) + \Gamma_{12}^{2}(q) b(q) u'(0) + \Gamma_{22}^2(q) b(q) v'(0) \right)\sigma_v(q). $

where $\pi_{T_pS}$ is the projection on $T_pS$ and $\Gamma_{ij}^{k}$ are the Christoffel symbols.

Writing the covariant derivative in this form is clear that this definition does not depend of the curve $\alpha$ chosen. But why this derivative does not depend on the parametrization $\sigma$? Can someone help me?

Answer 1

The way the Christoffel symbols change when you change parametrizations is not something you want to mess with here. The key point is the definition of the covariant derivative: As you wrote, it is the projection onto the tangent plane of $S$ at $\alpha(0)=p$ of the derivative of the vector field $\omega$ along $\alpha(t)$ at $t=0$. Neither of those entities depends on the parametrization $\sigma$ of $S$. (I would recommend making the projection more explicit: I would write $\pi_{T_pS}$ rather than just $\pi$.)