1
$\begingroup$

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?

$\endgroup$

2 Answers 2

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Likely, the formulation in the question was wrong. The real statement should say about deformation of a surface which moves the asymptotic line to a principle line (or something like that). $\endgroup$ Jul 14, 2017 at 16:05
  • $\begingroup$ You are right @Robert. In this way is trivial and what we are really looking for is a technique! We are studying flecnodal curves in a surface and we need a transformation which sends flecnodal curves into ridge curves (if it exists!). Do you know any reference about? $\endgroup$
    – Tomás
    Jul 14, 2017 at 16:44
4
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.