# Why the covariant derivative does not depend of the parametrization?

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?

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 $\pi$.)
• But $a$ and $b$ depends of the parametrization $\sigma$, how can I affirm that $\frac{d \omega (\alpha(t) )}{dt}$ doesn't depend of the chosen parametrization? (I changed $\pi$ for $\pi_{T_pS}$ as you suggested) Oct 8, 2017 at 20:46
• You can figure out how $a$ and $b$ change, but the real headache is how the Christoffel symbols change. They are not tensors, so there's a real mess. You start with a conceptual definition (as I gave it) and then deduce a formula in a particular parametrization; since you started with something well-defined (projection of a vector onto a $2$-dimensional subspace), you know the answer has to be independent of parametrization. Oct 8, 2017 at 21:07