Transformation which sends asymptotic lines to principal lines over a surface

Question

Suppose $M$ is an $C^\infty$ surface in $\mathbb{R}^3$ and let

$$ K_-=\{p\in M:\ K(p)<0\}\neq \emptyset, $$

where $K$ denotes Gaussian curvature. Consider the following statement:

Let $p\in K_-$. Then, there exists a neighborhood $U$ of $p$ and a diffeomorphism $T:U\to U$ such that if $J$ is an open interval and $\gamma:J\to U$ is a asymptotic line then $T(\gamma):J\to U$ is a principal line.

My question is the following: I can't remember in what paper I saw this result. I only remember that one of the authors name was Lie. Does anyone knows where I can find it?

Answer 1

I don't know a reference, but, as you've stated it, this is a trivial result:

If $p$ is a point on a smooth surface $S\subset\mathbb{R}^3$ at which the Gauss curvature is negative, then $p$ is non-umbilic, so, in an open $p$-neighborhood in $S$, both the principal curves and the asymptotic curves define transverse foliations of $S$. In particular, there exist $p$-centered local coordinates $(u,v):V\to\mathbb{R}^2$ such that the $u$-level curves and $v$-level curves are the principal curves and there exist $p$-centered local coordinates $(x,y):V\to\mathbb{R}^2$ such that the $x$-level curves and the $y$-level curves are the asymptotic curves. Hence there is an open $p$-neighborhood $U\subset V$ on which there exists a (unique) diffeomorphism $T:V\to U$ fixing $p$ such that $x = u\circ T$ and $y = v\circ T$. Since $T$ takes each $x$-level curve to a $u$-level curve and each $y$-level curve to a $v$-level curve, it carries every asymptotic curve in $V$ to a principal curve.

I suspect that you may be remembering a paper in which some specific method is given for constructing such a $T$ with more particular properties than just taking asymptotic curves to principal curves, but I confess that I don't remember seeing a discussion about such a particular construction in any of the classical literature that I have read.

Answer 2

This sounds a lot like Lie's line-sphere transformation, first alluded to in a joint paper with Klein, perhaps his only official coauthor ever (1870, §7). As summarized by Darboux (1871):

At the end of the Note (...) the authors obtain, by a transformation method which maps points of a line to rectilinear generators of a sphere, this important theorem: Whenever one knows the lines of curvature of a surface, one can deduce the asymptotic lines of another surface.

Note: another surface — I am not sure the transformation can be seen as going from $M$ to itself. Klein-Lie certainly didn't phrase the theorem as you do, and I don't know who might have: Helgason (1994, pp. 6-9), while giving a modern account and historical remarks (the transformation is an expression of the exceptional isogeny $\smash{\mathrm{SL}_4\to\mathrm{SO}_6}$), also notes the dearth of textbook expositions since Blaschke (1929).