Let $X \subseteq \mathbb{R}^n$, $A \subseteq \mathbb{R}^p$ be open sets. Suppose that there is an injective $C^\infty$ map $\varphi : A \to X$. Furthermore, suppose that we have a $C^\infty$ map $F : X \times A \to \mathbb{R}^n$.
Let $DF_X(x,a)$ denote the Jacobian of $F$ with respect to the $X$-variables at the point $(x, a) \in X \times A$. Let $I_n$ denote the $n \times n$ identity matrix. Suppose $F$ satisfies the following conditions.
- $F(x, a) = R(x,a)(x - \varphi(a))$ for a smooth matrix-valued function $R(x,a) \in C^\infty(X \times A ; \mathbb{R}^{n \times n})$, with $R(\varphi(a), a) = I_n$ (in particular, this means $F(x,a) = x - \varphi(a) + O(|x - \varphi(a)|^2)$.
- $DF_X(\varphi(a), a) = I_{n}$, each $a \in A$ (in fact condition 1 implies condition 2).
I would like to show that for each $a_0 \in A$ fixed, there exists two open sets $X_0 \subseteq X$, $a_0 \in A_0 \subseteq A$ such that the map $X_0 \ni x \mapsto F(x, a)$ is a diffeomorphism for each $a \in A_0$.
Context: This kind of construction appears after the proof of a Morse Lemma is Duistermaat's Fourier Integral operators (page 13). The ultimate goal is to study the behavior of oscillatory integrals of the form $$\int e^{itf(x,a)}g(x,a,t)dx,$$ near where $\nabla_x f(x,a) = 0$.
My attempt at a solution: The determinant $\det(DF_X)$ is a continous function of $x$ and $a$. Because $DF_X(\varphi(a_0), a_0) = I_{n}$ there are open sets $X_1 \ni \varphi(x_0)$ and $A_1 \ni a_0$ such that $\det(DF_X) \neq 0$ everywhere on $X_1 \times A_1$. It remains to show that $X_1$ and $A_1$ can be shrunk in a suitable way to $X_0 \subseteq X_1$, $A_0 \subseteq A_1$ so that for each $a \in A_0$, $X_0 \ni x \mapsto F(x,a)$ is injective (here, we are appealing to the well-known fact that an injective smooth map with nonsingular Jacobian everywhere is a diffeomorphism).
However, I am stuck trying to show this kind of injectivity. By the inverse function theorem, for each $a$, I can get an open set $X_a \ni \varphi(a)$ depending on $a$ such that $X_a \ni x \mapsto F(x, \varphi(a))$ is injective (in fact $F(\cdot, a)$ is a diffeomorphism there). But I am still seeking this "uniform injectivity" where the domain $X_a$ can be made independent of $a$ for $a$ ranging in an open set.
Hints or solutions are greatly appreciated!
For an $n \times n$ matrix $M = (m_{ij})$, let $\| M\|$ denote the Frobenius norm or $L^2$-norm of $M$: $$\|M \|^2 = \sum_{i,j = 1}^n m_{ij}^2.$$
The maps $$X \times A \ni (x,a) \mapsto DF_X(x,a) \in \mathbb{R}^{n \times n},$$ $$X \times A \ni (x,a) \mapsto \det(DF_X(x,a)) \in \mathbb{R},$$
are continuous function of $x$ and $a$ because $F \in C^\infty$. Since $DF_X(\varphi(a_0), a_0) = I_{n}$, continuity gives us open sets $X_0 \subseteq X$, $A_0 \subseteq A$ such that $(\varphi(a_0), a_0) \in X_0 \times A_0$ and for all $(x,a) \in X_0 \times A_0$, we have $$\|I_n - DF_X(x,a) \| < \frac{1}{2},$$ $$\det(DF_X(x,a)) \neq 0.$$
We can even shrink $X_0$ to be a Euclidean $n$-ball about $\varphi(a_0)$ if we like, and we will do so because it has the nice property of being convex (we will use this below).
We will show that for each $a \in A_0$, the map $ X_0 \ni x \mapsto F(x,a)$ is injective. This will complete the proof because any injective smooth map whose Jacobian is nonsingular everywhere is a diffeomorphism onto it's image (a reference for this in Corollary C.36 of Lee's Book Introduction to Smooth Manifolds).
To show injectivity. Fix $a \in A_0$. Suppose we have $x, x' \in X_0$ with $F(x,a) = F(x',a)$ yet $x \neq x'$. Then the following two equations hold $$x = x -F(x,a) + F(x,a),$$ $$x' = x' - F(x',a) + F(x,a).$$ Subtracting the two equations and taking norms gives: $$ | x - x'| = |(x - F(x,a)) - (x' - F(x',a))| \le \sup_{\overline{x} \in [x,x']}\|I_n - DF_X(\overline{x}, a) \||x - x'| \\ \le \frac{1}{2} |x - x'|.$$ Here, $[x,x']$ denotes the line segment in $X_0$ joining $x$ and $x'$ (such a line segment exists because we assume $X_0$ is a ball, and we have invoked the mean value inequality for vector-valued functions.
This is a contradiction, and so we have injectivity as desired.