Do I understand metric tensor correctly? So I've been studying vectors and tensors, and I'm trying to understand metric tensors.
As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors regardless of whether their basis has changed.
So if you had a set of vectors in say a Cartesian coordinate space, using a metric tensor to describe the distances between the vectors even if the primed basis describes them in Spherical coordinates, are he same distances? This isn't becoming quite intuitive to me yet.
This equation might clarify where I understand the definition of a metric tensor:
$ds^2= g_{11} dx^2 + g_{22} dy^2$

@Anon's answer focuses on the distance function, but perhaps one could relate to the question as stated, in the context of the metric tensor.
@mathacka: your notation for the metric tensor is highly unusual.  Which book are you using?  Usually what one writes down is an expression of the sort $ds^2= g_{11} dx^2 + g_{22} dy^2$ (in the simplest diagonal case).  To be more fancy, one could set $du^1=dx$ and $du^2=dy$, and write $ds^2 = g_{ij} du^i du^j$ with the implied summation over both indices (here $g_{12}=0$ in the diagonal case). This last expression looks somewhat reminiscient of what you wrote.  More details can be found in my course notes at http://u.cs.biu.ac.il/~katzmik/88-526.html
To answer the follow-up question, one key use of the $\Gamma_{ij}^k$ symbols is in writing down the geodesic equation: ${\alpha^k}^{\prime\prime}+\Gamma_{ij}^k$ $ {\alpha^i}^{\prime}{\alpha^j}^{\prime}=0$.
