Let $M$ be a smooth manifold with an affine connection $\nabla$. A vector field along a curve $c$ is called parallel if its covariant derivative $\frac{DV}{dt}$ along $c$ is equal to $0$.
This notion must have something to do with the intuitive notion of parallelism in the Euclidean space. But what is this relationship? A concrete example of a parallel vector field in the context of the most natural example of Riemannian manifolds, namely $\mathbb R^3$ will illustrate this notion.
Edit: what I have in mind is something like the following: For example consider the unit circle $t\rightarrow \exp(it)$. Consider its velocity field which is $t\rightarrow i\exp(it)$. After a boring calculation of its covariant derivative, does it follow that the covariant derivative equals $0$?
Best Answer
The relation is as follows. Consider the case $M=\mathbb{R}^n$, where $\nabla$ is the standard connection. Namely, $\nabla$ is the usual derivative. Let $\gamma:I\to\mathbb{R}^n$ be a regular path, and let $X$ be a vector field along $\gamma$. Then $X$ is parallel, in the sense that its covariant derivative vanishes, if and only if its values along $\gamma$ are parallel to one another, in the more naive sense.