Levi-Civita connection

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Can any one help me on this question:
Under what relation between vector fields X and Y, the Levi-Civita connection of X with respect to Y, \nabla_{Y}X is 0?
Any answers or suggestion will be highly appreciated.
 
243
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What would be the geometric interpretation of [tex]\nabla_{Y}X = 0[/tex]?
 
489
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The geometric interpretation is as follows: think of the flow lines of Y as paths in the manifold. What this condition is saying is that X does not change along these flow lines (with respect to the connection, or equivalently with respect to the metric in the case of the L-C connection). This is the idea of parallel transport, which is a very geometric concept. Given a curve c(t) and a vector at c(0), there is a unique extension of that vector to a parallel vector field along the curve.

It might be helpful to draw some curves in the plane and find out how a parallel vector field must behave.
 

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