Let's say that I know the values to $\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\in\mathbb{C}$ along some curve parameterized by real parameter $t$. If at all, how can I use this information to recover the general vector field $\mathbf{F}(\mathbf{r})$ that does not depend on the direction of the tangential vector $\mathbf{r}'(t)$ at each point along the curve? Or, what information can this data give me about the topology of the underlying base space?
Can I can use $\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)$ to define a vector field along it (a complex number may be thought as a vector with components comprising the real and imaginary parts)? If so, can I extend this field to a unique vector field over the entire $\mathbf{r}$ space to recover information about the 2-dimensional manifold it lives in? If I can, under what conditions can I do so? I am only interested in the case where $\mathbf{r}$ are coordinates of some smooth 2-dimensional manifold.
I am coming from a physics background, and am not sure where exactly to look, nor am I sure what mathematical language I should use to tackle this problem. Some initial thoughts are that I can define a vector field along the curve, and then define a covariant derivative field, and then extend it beyond the curve under some conditions (for example by considering a tubular neighborhood along it). At first I thought analyzing the 2nd order tangent space might help, but it was too abstract for me.
I saw these similar posts, but am not sure how they answer my question directly:
Definition of vector field along a curve
Covariant derivative of a vector field along a curve
