Question

There are quite a few german mathematical theorems or notions which usually are not translated into other languages. For example,

Nullstellensatz, Hauptvermutung, Freiheitssatz, Eigenvector (the "Eigen" part), Verschiebung.

For me, as a German, this is quite entertaining. Do you know other examples? Please one per answer, please give a reference for the term or a short explanation of what it means.

It would be great to see an explanation why there is no translation.

EDIT: Some more examples can be found at Wikipedia: Ansatz, Entscheidungsproblem, Grossencharakter, Hauptmodul, Möbius band, quadratfrei, Stützgerade, Vierergruppe, Nebentype.

Answer 1

And what about the Wiedersehen metric?

Answer 2

Spiegelungssatz. The meaning of this theorem is briefly discussed in the article: Iwasawa theory and $p$-adic deformations of motives [MR1265554 (95i:11053)] by Ralph Greenberg.

Let $p$ be an odd prime, and $K_\infty=\mathbf{Q}(\mu_{p^\infty})$. Let $L_\infty$ denote the maximal unramified abelian pro-$p$ extension of $K_\infty$, and $M_\infty$ the maximal abelian pro-$p$-extension of $K_\infty$ that is unramified outside the primes above $p$. Let $Y_\infty={\rm Gal}(L_\infty/K_\infty)$ and $X_\infty={\rm Gal}(M_\infty/K_\infty)$. We can decompose ${\rm Gal}(K_\infty/\mathbb{Q})\cong\Delta\times\Gamma$, where $\Delta={\rm Gal}(\mathbf{Q}(\mu_p)/\mathbf{Q})$ and $\Gamma\cong\mathbf{Z_p}$. Both $Y_\infty$ and $X_\infty$ have a natural structure of $\Lambda$-modules ($\Lambda=\mathbf{Z_p}[[\Gamma]]$) coming from the action of ${\rm Gal}(K_\infty/\mathbf{Q})$ by inner automorphisms. The latter action gives in particular an action of $\Delta$, and hence we can decompose $Y_\infty=\bigoplus_{i=0}^{p-2}Y_\infty^{\omega^i}$ and $X_\infty=\bigoplus_{j=0}^{p-2}X_\infty^{\omega^j}$ as $\Lambda$-modules, where the superscript denotes isotypical component under the action of $\Delta$, and $\omega:\Delta\rightarrow\mu_{p-1}$ denotes the mod $p$ cyclotomic character. The spliegelungsatz is then described by Greenberg in loc. cit. as an argument using Kummer theory and class field theory that allows to relate the structures of $X_\infty^{\omega^j}$ and $Y_\infty^{\omega^i}$ for $i+j\equiv 1\pmod{p-1}$ as $\Lambda$-modules.

Answer 3

The practice to use Gothic letters sometimes for ideals ($\mathfrak{a}$, $\mathfrak{b}$, ...) and often for Lie algebras ($\mathfrak{g}$, $\mathfrak{h}$, ..) seems to be of German origin.

Also to use the lesser known "kernel" instead of the better known "core" seems to stem from the German "Kern".

Answer 4

The following theorem is known as Kugelsatz:

Let $X$ be an open set in $\mathbb{C}^n, \quad n \geq 2$ and $K \subset X$ a compact subset such that $X\setminus K$ is connected. Then the restriction map $\rho: \mathcal{O}(X) \mapsto \mathcal{O}(X \setminus K)$ is an isomprphism of $\mathbb{C}$-algebras (this version after: Volker Scheideman, Introduction to Complex Analysis in Several Variables, Birkh\"{a}user 2005).

The first result of this kind is due to Hartogs, with $X$ and $K$ being concentric euclidean balls, hence the name (Kugel=ball). many texts in several complex variables have been written by German-speaking authors (Grauert+Fritzsche, Kaup brothers are other examples), so the German name stuck even in the English version. The theorem is also referred to as "tomato can principle".

Answer 5

Zusammenstellung. Means "compilation" or "survey". Can be used in the first section of a paper, as one starts compiling "preliminary facts" to refer to later in the paper. That's the way I've seen it used in a paper by Raoul Bott.

Answer 6

There's a kind of combinatorial design called a gerechte design - essentially it's a Latin square with additional block constraints. (I gather there's been a fad in recent years for newspapers to print partial gerechte designs of a certain kind for readers to complete.) As a technical term, the word comes from the following paper:

W. U. Behrens (1956). Feldversuchsanordnungen mit verbessertem Ausgleich der Bodenunterschiede. Zeitschrift für Landwirtschaftliches Versuchs- und Untersuchungswesen, 2, 176–193.

Behrens' gerechte designs were 'fair' in how they apportioned plots of land to different treatments in an agricultural trial.

Answer 7

If you think of the symbols, you can also see Gothic, alternatively called German, letters. Also, in algebraic topology, it is common to show the cycles by $Z$, which is the first letter of Zykel.

Also, many words that are Latin or Greek, in terms of the ingredients, were first coined and used in German, like Topologie which used to be called Analyse Situs.

It was common to show curvature by $K$, which stands for Krummung. Also, it was common to show a domain by B, for Bereiche. Or in riemannian geometry, the metric tensor is represented by $g$, which stands for Gravit\"at Also, Faltung used to be common in English before the word convolution took over.

I can also add Umlaufssatz in the differential geometry of surfaces.

There are so many more...

Answer 8

I would like to mention a handful of examples that may be considered passé nowadays, but were prominent at some point in time.

• schlicht: I dare to address this one again because I consider that the feedback in the comments below Gottfried's entry is kind of misleading. About this one, Boas says that (see [1, page 97]):

«... When I was an undergraduate, there was no regular colloquium Harvard, but there was a Mathematical Club, whose meeting were regularly attended by faculty. Once somebody gave a talk on schlicht functions. After the talk, Julian Lowell Coolidge asked plaintively whether there was an English word for 'schlicht'. Osgood replied, "Well, you could call them univalent functions, and everybody would know that you meant 'schlicht'". You need to know that Osgood had been trained in Germany, wrote his treatise on complex analysis in German, and was apt to tell German jokes to his classes. »

• limes: That's right... It was not a typo in Ahlfors's text on Complex Analysis. I recently came across this one in another book, but I just can't recall which one it was.

• eine Drehstreckung: Tristan Needham recalls this one when he apologizes for the coinage of the term 'amplitwist'. More specifically, he writes

«... To the expert reader I would like to apologize for having invented the word 'amplitwist' ... as a synonym (more or less) for 'derivative', as well the component terms 'amplification' and 'twist'. I can only say that the need for some such terminology was forced on me in the classroom: if you try teaching the ideas in this book without using such language, I think you will quickly discover what I mean! Incidentally, a precedence argument in defence (sic) of 'amplitwist' might be that a similar term was coined by the older German school of Klein, Bieberbach, et al. They spoke of 'eine Dhrestreckung', from 'drehen' (to twist) and 'strecken' (to stretch). »

Last but not least, in several works of old (z.B., Perron's Die Lehre von den Kettenbrüchen, Knopp's Theory and Application of Infinite series, Khinchin's Continued Fractions), there appears the following notation for general continued fractions:

$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$

Guess what the $\mathrm{K}$ stands for...

References

[1] Lion Hunting & Other Mathematical Pursuits: A Collection of Mathematics, Verse and Stories by Ralph P. Boas Jr.

Answer 9

Ganzstellensatz.

Answer 10

Bew (short for beweisbar, introduced by Gödel's incompleteness paper) is still used as a provability predicate in some mathematical logic papers.

In physics and other subjects (not so much in math) we hear about plenty of Gedankenexperiments.

Don't forget Hilbert's Satz 90, anomalous because of the "90" and not just the "Satz".

There are also French words like étale cohomology.

Answer 11

There is Ahlfor's scheibensatz in complex function theory, which is a generalization of Ahlfors five islands theorem

Answer 12

I believe Albrecht Frölich uses the german term beweis, instead of the english proof, in his chapter of the classic "Algebraic number theory". (EDIT: In my original version, I translated beweis to example. I shouldn't trust my poor knowledge of German... )

Answer 13

Einheit = word for unit in algebra. Hence, some use the notation $e\in G$ to denote the element of a group such that $ex = xe = x , \forall x \in G$. Unit is the appropriate translation, yet some algebraist still use the letter $e$ to denote the identity element in a group.

Answer 14

In "Functional Analysis" by Kosaku Yosida he denotes the closure of a set $M$ by $M^a$. He explains that it is a shortcut from German abgeschlossene Hulle.

Answer 15

There is also the Quermassintegral (mixed volumes of the form $V(K,K,\ldots,B,B)$ where $B$ is the unit ball, see Wikipedia), which I'm not even sure is German (not a lot of Qs in German usually).

Answer 16

In topology the separation axioms $T_0$ , $T_1$ .. etc, where the $T$ stands for Trennungsaxiom

Answer 17

One that is similar in spirit "eigenvalue" in that it mixes the two languages is $$ \text{umkehr map} $$

Answer 18

It's early in the morning, so maybe I missed it in the answers above, but, if we're including symbols, then the obvious example is $\mathbbm{Z}$, the integers, or zahlen!

Ooops! It is early in the morning... I see that Roland noted that the symbol for the integers (which I also can't seem to get to process properly) just a few comments above.

Answer 19

In Swedish, a field is called a 'kropp', a body. This of course from the German word Körper.

Answer 20

Schubfachprinzip ("drawer principle" or "shelf principle" or "Dirichlet's box principle"). It is now easy to guess we are talking about P-H P.

Answer 21

"schlichtartig" refers to a surface on which every simple closed curve which separates locally, also separates globally. Hence it means roughly "planar". This is used in the conformal mapping theory of Riemann surfaces. Introduction to Riemann Surfaces, Springer, p. 91. I know only a little German but it seems to translate something like "simply behaved"?

Answer 22

Anzahl-theorems is one I have recently read in Wan's book on classical groups.

Answer 23

The $\int$ symbol is a german S introduced by Leibniz and stands for Summe (Sum)