Parallel Transport and covariant derivative I have been trying to understand the notion of parallel transport and covariant derivative.
I am unable to see why the change in a vector when it is parallel transported from one point to another shouldn't be a vector. If it is, why isn't the Levi-Cevita connection not a tensor ? 
Hence my questions are :
What is a connection geometrically ? What is parallel transport in a particular coordinate system geometrically ?
 A: In general the base vectors ($\vec{e}_{i}$) are not constant, e.g., in polar coordinates the radial vector do not point in the same direction (unlike the Cartesian base vectors $\hat{i},\hat{j},\hat{k}$).
If one takes the j-th base vector $\vec{e}_{j}$ and consider its change if one moves on the direction defined by the i-th vector, mathematically this is $$\partial_{i}\vec{e}_{j} \,\, .$$
Since the result must be a vector (the changed vector), the new vector can be expressed as a linear combination of the vector basis... say $\xi^k \,\vec{e}_{k}$ for certain values of $\xi^k$. However, the values of $\xi^k$ depend on the choice of the i-th and j-th *directions.
Thus, one usually expresses it as $$\partial_{i}\vec{e}_{j} = \Gamma_{ij}{}^k\,\vec{e}_{k} \,\, .$$
The connection encodes the information about how the vector basis change.

As I mention before, the Cartesian vector basis are constant. Therefore, any derivative of any Cartesian vector basis vanish... i.e., all the possible $\Gamma$'s are zero!!!

For transporting a vector, you have to take into account how the basis change.
The concept of parallel transport is to transport the vector in a way that it does not change with respect to the moving basis.
A: The connection relates vectors at two different places, each of which has its  own basis set. When you change basis you can make different changes at the two points, and so the change in the connection has to know about two different transfomations. If it were a vector or tensor it would only have to know about the basis transformation  at  one point. Consequently the array of numbers specifying the connection cannot be a tensor.
