I am having trouble proving this, it seem obvios to me but i dont know how to prove it:
Let $M$ be a surface embedded in $\mathbb{R}^n$ with the induced metric and let $N_1$ be a sub-bundle of the normal bundle that is parallel in the normal bundle, with that i mean the normal connection vanishes $\nabla_X ^\bot \eta = 0 $ for every section $\eta \in N_1$ and every $X$ tangent to $M$.
Then i take and arbitrary vector $\eta_p \in (N_1)_p$ where $p$ is an arbitrary point of $M$ and consider the parallel transport of $\mathbb{R}^n$ of $\eta_p$ along a curve in $\gamma$ in $M$ .
Is true that $\eta(t) \in N_1(\gamma(t)) $ ?