I am recently concerned with the following problem: Given a parametrized curve in $\mathbb{R}^6$, what is the condition that it belongs to a hyper-sphere of dimension 5?

The more general question would be what are the conditions for a given parametrized curve in $\mathbb{R}^n$ to belong to a given hyper-surface of $\mathbb{R}^n$? The easiest answer is the existence of an isometric embedding. But I think that is a question of integrability. I was trying to see the simplest case: the condition of a space curve $\mathbf{r}(s)$ in $\mathbb{R}^3$ to belong to a given surface $\mathbf{R}(\theta^1,\theta^2)\in\mathbb{R}^3$. Intuitively, it simply means that at every point on the curve, the vector space formed by the tangent vector and either the normal vector or the binormal vector merges with the tangent space at the corresponding point of the surface. But there has to be an intrinsic way of saying the same thing, namely ,for example, a compatibility between the Serret-Frenet curvature-torsion pair of the curve and the curvature tensor of the surface. Either of the two choices of vector spaces mentioned above gives rise to a 2 dimensional differentiable distribution in $\mathbb{R}^3$ along the given curve. For an integrable surface to exist, this distribution has to be involutive. Then, we can define an isometric diffeomorphism between this surface and then given one.