Suppose a tangent vector field is given on a planar curve and one asks the following question: What is the condition on this tangent vector field that it comes from the ordinary Euclidean projection of a constant vector field defined on a planar 2 dimensional neighbourhood of this curve? A simple necessary and sufficient condition is if one parallelly translates, in the usual Euclidean sense, each tangential vector of the given field to a common point on the plane, the tips of the translated vectors describe a circular arc. This can be easily proven by elementary Euclidean geometry.

My question is if it can be seen in a more matured way, I mean, in terms of parallelism, connection, covariant derivatives, in these languages. Specifically, what I am looking for is some specially defined connection with respect to which the covariant derivative of the given tangential field being zero implies, and is implied by, the field be a projection of a constant field in the embedding space.

Sorry for this late amendments which I should have pointed out earlier. There are the following general versions of this problem:

**Generalization to Surfaces in $\mathbb{R}^3$**

Suppose we have an oriented smooth surface $S$ embedded in $\mathbb{R}^3$ and a sufficiently smooth tangent vector field $\mathbf{w}$ defined over $S$. Then it can be shown, using elementary analysis, that the vector field $\mathbf{w}$ comes from the Euclidean projection of a constant vector field $\mathbf{C}$ defined over a tubular neighbourhood of $S$ in $\mathbb{R}^3$, if, and only if, the surface described by the vectors in the given field $\mathbf{w}$, when translated parallely to the origin (let's take the common point as the origin), is part of a 2-sphere in $\mathbb{R}^3$ that passes through the origin. In fact, $\mathbf{w}$ satisfies the following equation:

$$(\mathbf{w}-\frac{\mathbf{C}}{2})\cdot(\mathbf{w}-\frac{\mathbf{C}}{2})=\frac{\mathbf{C}\cdot\mathbf{C}}{4}.$$

**The most general version**

Let $S$ be a $p$-dimensional sufficiently smooth submanifold embedded in $\mathbb{R}^n$ and $\mathbf{w}$ be a tangential vector field defined on $S$. Then, $\mathbf{w}$ comes from the Euclidean projection of a constant vector field $\mathbf{C}$ defined on a neighbourhood of $S$ in $\mathbb{R}^n$ if and only if the submanifold described by the vectors in the given field $\mathbf{w}$ when translated to a common origin is part of an $(n-1)$-dimensional sphere passing through the origin.

I am looking for an interpretation of these general results in terms of connection on some appropriately defined bundle over the embedded submanifold $S$.

**Note:** We only are given with the tangential component of the field and know nothing about the transverse component. The above results, also, involve only the tangential component.