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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 ?

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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 is a way that it does not change with respect to the moving basis.

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