In my intro to differential geometry course the following was said on parallel transports.
We define the rate of change of a tangent vector $$v = v^a \frac{\partial}{\partial x^a} \in T_p(\mathbb{R}^n)$$ along a curve $\lambda : I \to \mathbb{R}^n$ in the following way. Note that $p = \lambda(t)$ for some $t \in I$. Consider another tangent vector $$X_p = X^a \frac{\partial}{\partial x^a} \in T_p(\mathbb{R}^n)$$
Then the rate of change of $v$ is defined as $$\left(\frac{Dv^a}{dt}\right)\bigg|_{\lambda(t)} = X^b v^{a}_{; \ b} = X^b\left(\frac{\partial v^a}{\partial x^b} + \Gamma^{a}_{b \ c} v^a\right)$$
and since $$X^b = \frac{dx^b(t)}{dt}$$ we have
$$\left(\frac{Dv^a}{dt}\right)\bigg|_{\lambda(t)} = \frac{dv^a}{dt} + \Gamma^{a}_{b \ c} v^c \frac{dx^b}{dt}$$
Now I have a few questions about the above
- Firstly is the above definition correct and rigorous?
- In order to define $\left(\frac{Dv^a}{dt}\right)\bigg|_{\lambda(t)}$ we needed to consider another tangent vector $X_p$, how do we know that $\left(\frac{Dv^a}{dt}\right)\bigg|_{\lambda(t)}$ doesn't depend on the choice of $X_p$?
- What exactly is $\left(\frac{Dv^a}{dt}\right)\bigg|_{\lambda(t)}$, it looks symbolically like a derivative and in it's definition the components of the covariant derivative of $v$ pop up, how are the two concepts related?
Just a comment, I think that the need to introduce another tangent vector $X_p$ is as a result of my lecturer wanting to avoid introducing vector fields