@GFauxPas I don't have any idea what "mental arithmetic" "trick" you're seeing there. It's super important to understand matrix multiplication both in terms of rows and in terms of columns.
short question: If I have a sequence of isometries $f_n$ on a manifold $M$ so that $f_n(x)$ converge for some point $x\in M$ and the differentials $D_x f_n$ converge, does there exist an isometry $f$ against which $f_n$ converge (uniformly on compacta)?
if $M$ is compact then yes by an argument with azerla ascoli, but im not finding easy counterexamples for noncompact $M$
@s.harp Interesting question. What if $f_n$ is the identity except on union of balls of radius $1/2$ a distance $1,2,\dots,n$ away from the origin? (Do a $90^\circ$ rotation on half the ball, and then "unrotate" to join it to the identity by the boundary of the big ball.) So, pointwise, $f_n$ converges to the identity, but clearly that won't be uniform on compacta.
The derivative of $f_n$ is a rotation at every point, so it's an infinitesimal isometry. This might need a little repairing, but ...
