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In Lee's "Riemannian Manifolds: An Introduction to Curvature" given a curve $\gamma:[a,b]\to M$ and a tangent vector $V_0\in T_{\gamma(t_0)}M$, where $t_0\in [a,b]$, there is a drawing of the parallel translate of $V_0$ in figure 4.7. This parallel vector field seems to be drawn so that it is perpendicular to $\gamma$ at every point.

Is there a geometric interpretation as to why parallel vector fields are called parallel? Why is the parallel translate drawn in this way?

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    $\begingroup$ Actually, the vector field in that figure is not meant to be perpendicular to the curve -- what I had in mind was that both the curve and the vector field lie in the (Euclidean) plane, and the vector field points in the same direction at every point of the curve. In Euclidean space, this means that all the vectors are parallel in the usual sense. On an abstract Riemannian manifold, "pointing in the same direction" has no meaning, but being parallel along a curve is the closest we can come to saying that the direction of V doesn't change as you move along the curve. $\endgroup$
    – Jack Lee
    Sep 19, 2014 at 18:09

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A vector field $X$ along a curve $\alpha$ is parallel if $$\nabla_TX=0$$ This equation means that the vector field $X$ does change along $\alpha$. Geometrically, all values of $X$ along $\alpha$ seems to be parallel.

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