Confusion about parallel transport

[pervect]

So the first term in this DE keeps track of how the vector is changing with respect to our tangent plane, and the second term transport us from one tangent plane to the next?

Yes. In flat space with a cartesian coordinate system, for instance, the connection coefficients are all zero, therefore the condition of parallel transport is that the vector components don't vary at all.

I'm assuming that the non-homogeneous version of this DE would be non-parallel transport then? Thanks.

I would say that if the equation isn't satisfied, the vector is not being parallel transported by the connection. I'm not quite sure if this is equivalent to what you meant. For instance if we imagine a Cartesian coordinate system where all the connection coefficients are zero, but the vector components varied (were not constant), then the vector would not be being parallel transported by the chosen connection.

 

[emily1986]

I would say that if the equation isn't satisfied, the vector is not being parallel transported by the connection. I'm not quite sure if this is equivalent to what you meant.

After thinking about it, my question didn't really make sense. Thanks for your help, everything is much clearer now.