This was a statement I came across in a paper on what are called Transnormal curves by Bernd Wegner.

**Two immersions $f_1$ and $f_2$ of a smooth manifold $M$ into $\mathbb{R}^n$ such that at each point $p \in M$, both immersions are connected by a parallel section of their respective normal bundles** iff **the normal planes to $f_1(M)$ and $f_2(M)$ coincide $\forall p \in M$ at $f_1(p)$ and $f_2(p)$**

How can one prove this fact??

Further the author claims that in the special case of $M $ being $\mathbb{R}$ or $\mathbb{S}$, given the curves $\alpha = f_1$ and $\beta = f_2$, this boils down to saying

$$\beta(t)= \alpha(t)+ \lambda N(t)$$ and $$\text{prn}(\nabla_{\alpha'(t)}\bf{N}) = 0$$ where $\bf{N}$ denotes the principal normal vector field along $\alpha$ and $prn$ denotes the orthogonal projection to the corresponding normal (vector) space of $\alpha$ and $\alpha'(t)$ is the tangent vector field, with $\lambda$ being a constant.

What does the above condition tell me exactly??The first one is quite clearly saying that $\beta$ is a translation of $\alpha$ along the normal to $\alpha$, but what about the second condition about the projection of the covariant derivative??