Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a submanifold. Take a tubular neighbourhood of $N$ in $W$ and let $N'\subset W$ be a normal deformation of $N$. (By a normal deformation I mean the image of a section in the normal bundle, which has image in the tube.)
The preimages $j^{-1}(N)\subseteq M$ and $j^{-1}(N')\subseteq M$ are again a submanifold. Is $j^{-1}(N')$ a normal defomation of $j^{-1}(N)$ if the normal deformation $N'$ of $N$ was small enough?