Suppose a manifold $M$ with a covariant derivative operator $\nabla$ defined on it. A vector field $X$ on $M$ is a parallel vector field along a curve $c$ if
$$ \nabla_{c'(t)} X=0 $$It is also said that the vector field is constant along the curve.
This is well defined since covariant derivative only depends on the value at a point in the first vector field (see linear connection#Definition as operator).
If the vector field $X(t)$ is only defined along the curve $c$, we consider the covariant derivative along a curve $D_c$. With this in mind we can say when a vector field defined along a curve $c$ is parallel along the curve: $D_c X=0$ for every $c(t)$.
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Author of the notes: Antonio J. Pan-Collantes
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