Can you explain the geometric interpretation of the Levi-Civita connection?

[forumfann]

Can anyone 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.

 

[aPhilosopher]

What would be the geometric interpretation of $$\nabla_{Y}X = 0$$?

 

[zhentil]

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.