Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $f:\mathbb R^d\to\mathbb R^{d-k}$ be differentiable. Assume $0$ is a regular value of $f$ and hence $M:=\{f=0\}$ is a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$.
Let $x^\ast\in\mathbb R^d$, $\phi$ be a $C^1$-diffeomorphism from an open neighborhood $V$ of $x^\ast$ in $M$ onto an open subset $T$ of $\mathbb R^k$ and $\varphi:=\phi^{-1}$.
In chapter 5.3 of Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach the author claims that $\varphi$ "describes a point $x\in V$ as a function of its projection onto the subspace $L:=T_{x^\ast}\:M$"; more precisely, $\varphi$ is claimed to be of the form (or at least can brought into the form) $$\varphi:T\to V\;,\;\;\;\xi\mapsto x^\ast+Q\begin{pmatrix}\xi\\\eta(\xi)\end{pmatrix}\tag1,$$ where $Q:=(q_1,\ldots,q_d)$ is the orthogonal matrix constructed from an orthonormal basis $(q_1,\ldots,q_k)$ of $T_x\:M$ and an orthonormal basis $(q_{k+1},\ldots,q_d)$ of $N_x\:M$ and $\eta\in C^1(T,\mathbb R^{d-k})$ with $\eta(0)=0$.
Question: How can we show this?
First of all, by replacing $\phi$ with $\phi-\phi(x^\ast)$, we may clearly assume that $\varphi(0)=x^\ast$.
Moreover, we know that $T_{\varphi(x)}\:M=\mathcal R({\rm D}\varphi(\xi))$ and $N_{\varphi(x)}\:M=\left(T_{\varphi(x)}\:M\right)^\perp=\mathcal N({\rm D}\varphi(\xi))$ for all $\xi\in T$.
But I still don't see why we can write $\varphi$ in the form of $(1)$; in particular, how we need to define $\eta$.