## 7-2-68. Felix Klein to H. Poincaré, English translation

Leipzig, 14 May 1882

Dear Sir!

In response to the letter that just arrived, I would like to explain
briefly how I employ “continuity”. Only in principle, of course,
since the detailed exposition – which will be quite troublesome to
write up – can in any case be qualified in many ways. I will confine
myself throughout to the case of unbranched $\eta$-functions of the
second kind, as I called them in my note. Here it is primarily a
question of proving that the two manifolds to be compared – on the
one hand, the set of systems of substitutions considered, and on the
other, the set of all existing Riemann surfaces – not only have the
same dimension ($6p-6$ real dimensions) but form *analytic*
manifolds with *analytic* boundaries (in the sense of the
terminology introduced by Weierstrass). By the first lemma stated in
my previous letter, these two manifolds are related in a
$(1-x)$-valued manner, where by the second lemma, $x$, for the
various parts of the second manifold, can take on only the values 0 or
1. Now this relation turns out to be *analytic* and, as this
again follows from both lemmas, an analytic relation *whose
functional determinant vanishes nowhere*. From this I deduce that
$x$ always has the value 1. Let there be a transition from a region
with $x=0$ to another with $x=1$, then, in view of the analyticity of
the correspondence, the first region would correspond to definite
points (those actually attained) of the other region, and then for
these points the functional determinant of the relation would have to
vanish, contrary to the supposition. So much for my proof.

Mr. Schwarz put forward a quite different proof, also based on
considerations of continuity, when I visited him recently (on April
11) in Göttingen. Although I haven’t obtained his permission to do so,
I thought I should tell you about it all the same. He imagines the
Riemann surface cut appropriately and then covered infinitely often
with the various sheets so connected along the cuts that a complete
surface results corresponding to a set of polygons placed side by side
in the plane. This complete surface, if one may so term an infinitely
extended surface (which still requires clarification), is, in the
case of an $\eta$-function of the second kind (the case first
considered by Schwarz), *simply connected and with simple
boundary curve*, and then it is a question of seeing whether such a
simply connected surface with simple boundary can be mapped in the
usual way onto the interior of a disc. Schwarz’s train of
thought is very beautiful, in any case.

You ask about offprints. I would, above all, not wish to trouble you about that, particularly since I can always procure of all your works with the sole exception of your Thèse. So if you can send me a few items (I don’t have any so far), they would be most welcome.

Have you perhaps had the chance to read Lie’s theory of transformation
groups? Lie always takes the parameter figuring in his groups to be
complex-valued; it would be interesting to see how his results extend to
the situation where one considers groups generated only by a *real*
iteration of certain $\infty$ small operations.

Some time ago Hermite sent me one of the lithographed lessons of his *Cours
d’analyse*. Would it be possible (subject to payment of course) to
obtain the whole set? This would be particularly well met, in light of
the aims I’m currently pursuing in my seminar.

As always, your most devoted,

F. Klein

PTrL. Translated by R. Burns from the original German (§ 4-47-23) in Saint-Gervais (2016), with minor revisions by S.A. Walter. See also the French translation (§ 7-2-41), and Poincaré’s reply (§ 4-47-24).

Time-stamp: "14.05.2021 01:35"

## References

- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-68. Felix Klein to H. Poincaré, English translation.