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The following lemma is a key for Implicit Function Theorem, given in Topology and Geometry by Bredon; but I am unable to see what the theorem is stating, what its conditions imply geometrically. I tried in small dimension case ($n=m=1$), but could not proceed.

Let $\xi\in\mathbb{R}^n$ and $\eta\in\mathbb{R}^m$ be given. Let $f:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}^m$ be $C^1$ and put $f=(f_1,\ldots, f_m)$. Assume that $f(\xi,\eta)=\eta$ and that all the following partial derivatives vanish at $(\xi,\eta)$: $$ \frac{\partial f_i}{\partial y_j}(\xi,\eta)=0 $$ where $x_1,\ldots, x_n$ are coordinates in $\mathbb{R}^n$ and $y_1,\ldots,y_n$ are in $\mathbb{R}^m$. Then there exists real numbers $a>0$ and $b>0$ such that there exists a unique function $\phi:A\rightarrow B$ where $A=\{x\in\mathbb{R}^n \,:\, |x-\xi|\le a\}$ and $B=\{y\in\mathbb{R}^m \,:\, |y-\eta|\le b\}$, such that $$\phi(\xi)=\eta \hskip5mm \mbox{ and } \hskip5mm \phi(x)=f(x,\phi(x)) \hskip5mm \forall x\in A.$$ Moreover $\phi$ is continuous.

Can one explain what the lemma is saying?


The function $f$ is taken as $f(\xi,\eta)=\eta$; this looks like a projection map, but it is not overall projection map (since, only at point $(\xi,\eta)$, value is second component $\eta$). So, I did not get the importance of this assumption.

Further, the $m$ partial derivatives of $m$ components of $f$ are taken $0$, so in the $Df(\xi,\eta)$, the last $m\times m$ part is fully $0$; When one tries to put condition on an $m\times m$ matrix, the basic conditions come are whether it is singular or not; but he is putting more than singular condition. I did not get reason behind this.

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1 Answer 1

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What the lemma is saying is that if $F(\xi, \eta) = \eta$ and $D_yF(\xi, \eta) = 0$, then the equation $$F(x, y) = y$$ defines $y$ implicitly as a continuous function $y = \phi(x)$ of $x$ for $x$ near $\xi$.

Really the lemma is just the implicit function theorem applied to $G(x, y) = F(x, y) - y$.

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