Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal. For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$ We assume $(\mathcal{P})$. Show that there exist an orthogonal matrix $A$ and an element $b$ of $\mathbb{R}^n$ such that, for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. One may interpret the relations $\alpha_{i,j,k} = 0$ using matrix products.
Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote
$$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that there exist an orthogonal matrix $A$ and an element $b$ of $\mathbb{R}^n$ such that, for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$.
One may interpret the relations $\alpha_{i,j,k} = 0$ using matrix products.