I recently watched Sean Carroll's YouTube series on "The Biggest Ideas in the Universe". In his Geometry and Topology video, he says that the connection in Riemannian geometry describes how a vector gets parallel transported through the geometry.
I looked more into it and found that the Christoffel symbols are an array of numbers that describe the metric connection which itself describes how the basis varies from point to point.
The formula for the Christoffel symbols of the first kind is $\Gamma_{kij} = \frac{\partial \vec{e_{i}}}{\partial x^j} \cdot \vec{e_k}$. I'm trying to understand this formula intuitively, specifically how it describes how the covariant basis varies from point to point.
It seems to describe the projection of the rate of change of $\vec{e_i}$ with respect to the $x^j$ direction onto $\vec{e_k}$. As I understand it, this means that $\frac{\partial \vec{e_{i}}}{\partial x^j}$ is a vector that represents the change in the covariant basis vector $\vec{e_i}$ when you move in the $x^j$ direction. This resulting derivative vector then has components/projections along each covariant basis direction $\vec{e_k}$. So the Christoffel symbol describes the components/projections of each of these derivative vectors along each of the covariant basis directions at various points.
Is this the correct intuition behind the Christoffel symbols or do I have a misconception somewhere?
Also if this intuition is correct, I couldn't seem to get similar intuition for the Christoffel symbols of the second kind. Their definition is $\Gamma ^k_{ij} = \frac{\partial \vec{e_i}}{\partial x^j} \cdot \vec{e^k}$.
But I don't know how to intuitively grasp the projection onto $\vec{e^k}$ because the contravariant basis vectors $\vec{e^k}$ are vectors orthogonal to the covariant basis directions right? I guess the symbols of the second kind would be the projections of the derivatives onto the various directions orthogonal to the covariant basis?