# Adapted expression to a submanifold for a smooth function

I am trying to prove the following:

Lemma. Let $$S$$ be an embedded smooth submanifold of a smooth manifold $$M$$, let $$f\in C^\infty(M)$$ be such that $$f|_S=0$$, and pick $$p\in S$$. There exists adapted local coordinates $$x^1,\dots,x^n$$ around $$p$$ of $$M$$ with respect to $$S$$, with domain $$U\subset M$$ (i.e., $$U\cap S=\{x^{s+1}=0,\dots,x^m=0\}$$), and smooth functions $$g_{s+1},\dots,g_m\in C^\infty(U)$$ such that $$f=\sum_{i=s+1}^m g_ix^i$$ at $$U$$.

How can one proceed?

• I just realized this is a special case of Hadamard's lemma. Commented Jul 23, 2023 at 13:40

$$\def\d{\mathrm{d}}$$Let $$(U,\varphi)$$ be an adapted chart of $$M$$ to $$S$$, centered at $$p$$, and such that $$\varphi(U)$$ is convex. Then in terms of the local coordinates we have \begin{align*} f(x)&=\sum_{i=s+1}^mf(x_1,\dots,x_j,0,\dots,0)-f(x_1,\dots,x_{j-1},0,\dots,0)\\ &=\sum_{i=s+1}^m\int_0^1\frac{\d}{\d t}f(x_1,\dots,tx_i,0\dots,0)\d t\\ &=\sum_{i=s+1}^m\int_0^1x_i\partial_if(x_1,\dots,tx_i,0\dots,0)\d t\\ &=\sum_{i=s+1}^m x_i\int_0^1\partial_if(x_1,\dots,tx_i,0\dots,0)\d t\\ &=\sum_{i=s+1}^m x_ig_i, \end{align*} where $$g_i=\int_0^1\partial_if(x_1,\dots,tx_i,0\dots,0)\d t,\qquad i=s+1,\dots,m. \quad\square$$