The generalized Gauss map $G:M^{n}\rightarrow G_{n,m}$ of a given immersion $f:M^{n}\rightarrow \mathbb{R}^{m}$ assigns to each point $x \in M^{n}$ the point $f_{*}(T_{x}M)$ in the Grassmannian of $n$-planes in $\mathbb{R}^{m}$.
My question is: If $f,g:M^{n}\rightarrow \mathbb{R}^{m}$ are immersions with the same generalized Gauss map, then there exists $\Phi \in \Gamma(\mbox{End}(TM))$ such that $$g_{*}=f_{*}\circ \Phi.$$
Thanks for your hint or answer.