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[tex]\frac{d}{d\lambda} V^\mu + \Gamma^\mu_{\sigma \rho} \frac{dx^\sigma}{d\lambda} V^\rho = 0[/tex]

This is basically what every book says. But what exactly does it mean?

1) if you have a vector (field), there is a specific set of paths, that - when you transport the vector along them - will leave the vector 'unchanged'. That means that there are certain paths that change the vector i.e. stretches it or turns it or whatever.

2) the equation describes how the vector (field) should change when you move it along a certain path (any path you choose) in order to keep it constant with respect to the connection.

I can see arguments for both interpretations and it's driving me crazy. ;)

In flat space space you can parallel transport a vector in any direction you want - there are no right or wrong paths. This speaks for 2).

A geodesic is a path that parallel transports a vector that is the tangent vector to the path itself. This speaks for 1) (i.e. for a vector there is a specific parallel transport path, for the tangent vector to the path this is the path itself)

If you have a sphere and parallel transport the vector

[tex]V = (1,0)[/tex]

around a circle of constant [tex]\theta[/tex] (altitude) the vector changes like

[tex]V(\theta,\phi) = \left(\cos(\phi \cos\theta), \frac{-1}{\sin \theta}\sin(\phi \cos\theta)\right)[/tex]

This speaks for 2) as you can clearly choose any path (any altitude (except the north/south pole of course)). It also shows the vector is not constant with respect to the path (the path parameter being [tex]$\lambda = \phi$[/tex]) as it changes when you go round.

So...which one is it? What point am I missing? ;)