Suppose we have an isometric immersion $f:(M^m,g)\to N^{n}$ and let $\{\nu_\alpha\}$ be an orthonormal basis for $TM^\perp$. If we "shift" the manifold in the directions of $\nu_\alpha$ by real (sufficiently small) numbers $z^\alpha$, that is, for every local chart $(U,\phi)$ for $M$ we define the chart $(U,\phi_z)$ for $M_z$ by $$\phi_z=\phi+z^\alpha\nu_\alpha,$$ can we say anything about the metric $g_z$ of $M_z$ as a function of $g$ and $z$?
When $N=\mathbb{R}^{m+1}$ there is a known formula stating that $$ (1)\quad\quad g_z=g(I-zA)^2 $$ where the above expression is in matrix notation and $A$ is the representative matrix of the second fundamental form of $f$. Is there any similar result in higher codimension?