# Concrete interpretation of parallelism of a vector field along a curve

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 $$0$$?

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.