I am interested in the diffeomorphisms $T : \mathbb{R}^n \to \mathbb{R}^n$ that preserve the Euclidian norm, i.e such that $T(x)^\top T(x) = x^\top x$ for all $x \in \mathbb{R}^n$.
Do we know how to parameterize such transformations? Can we say that $T(x)$ has the form $T(x) = R(x) x$ with $R(x):\mathbb{R}^n \to \mathbb{R}^{n \times n}$ st $R(x) \in GL(n)$?
@KCd, that seems related to your answer Isometry without injection and surjection and your note https://kconrad.math.uconn.edu/blurbs/grouptheory/isometryRn.pdf