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Question:

Is there any criterion for using Riemann or Riemannian in terminology?


For example, Riemann occurs in these phrases:

Riemann surface, Riemann curvature tensor, Riemann-integrable, Riemann sum, Riemann sphere, Riemann-Roch theorem, Riemann Hypothesis...

... while Riemannian occurs in these phrases:

Riemannian geometry, Riemannian manifold, Riemannian measure, Riemannian metric...

Especially, why it's Riemann surface but Riemannian manifold?


Related links:

Why is “abelian” infrequently capitalized?

Mathematical adjectives that bear famous mathematician's names

Mathematical concepts named after mathematicians

Mathematician's names in structures.

How mathematical theorems and concepts gain their names?


Edit:

$1.$ From the comments, I realize it partly follows from conventions and habbits, though it may cause trouble when searching and might not be very friendly to non-English users. For example:

English: Riemann Surface, Riemannian Geometry (inconsistent)

French: surface de Riemann, Géométrie riemannienne (inconsistent)

German: Riemannsche Fläche, Riemannsche Geometrie (consistent)

Japanese: リーマン面, リーマン幾何学 (consistent)

Chinese: 黎曼曲面, 黎曼几何 (consistent)

$2.$ As the answer mentioned, when it's related to some definitions or properties, it's often "-ian", e.g. Riemannian metric, Gaussian process, Artinian ring, Noetherian ring.

$3.$ There're some typos about "Riemann" in this website.

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  • $\begingroup$ This is because English is a flexible language :) In German, Riemann surface is "Riemannsche Fläche", consistent with Riemannian manifold, "Riemannsche Mannigfaltigkeit". But note that "Abelsche Gruppe" also in English is never "Abel group", but Abelian group or abelian group. $\endgroup$ Feb 26, 2019 at 10:39
  • $\begingroup$ Has showed by the many links in yoyr post, natural language is not always "logical"; habits and conventions rule. $\endgroup$ Feb 26, 2019 at 10:40
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    $\begingroup$ I cannot resist to make a remark here. On this site "Riemann surface" is often misspelled as "Reimann surface", e.g., title of this post, or this post second line of this post, etc. $\endgroup$ Feb 26, 2019 at 10:50
  • $\begingroup$ And of course Cauchy-Reimann as well, reimann-integrable, Reimann Mapping Theorem, Lower Reimann Integral etc. $\endgroup$ Feb 26, 2019 at 10:53
  • $\begingroup$ French isn't consistent either, but it seems to be consistent with English : we have "surface de Riemann" but "variété Riemanienne" for instance $\endgroup$ Feb 26, 2019 at 11:48

1 Answer 1

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This is largely an English/linguistics question, but it requires some math knowledge to answer. I disagree with the idea that this is only because of the flexibility of English in using nouns as adjectives. For most of the examples listed in the OP, there is a difference between the two lists.

As a litmus test, if you would want to say "non-Riemann(ian)" then it has to be "Riemannian" since "non-Riemann" sounds weird to a native English speaker (cf. abelian group vs. Lebesgue integral).

Type 1:

The Riemann integral is one of many integrals, and it's the one associated with Riemann. We wouldn't usually look at a bunch of integrals and say "these 3 are Riemann(ian) and these 4 are non-Riemann(ian)"

A Riemann surface is a special sort of object, and one wouldn't have occasion to say "sure it's a surface, but this one is not Riemann(ian)".

The Riemann Hypothesis is a particular thing. We do not sort hypotheses/conjectures into the Riemann ones and the non-Riemann ones.

Etc.

Type 2:

Riemannian geometry is distinguished from other geometric structure in (differential) geometry. There is a book called non-Riemannian Geometry for this reason.

A metric on a manifold may or may not be positive definite, so we can ask whether a metric is Riemannian or not ("non-Riemannian metric" is rare, but would be understood).

Smooth manifolds can always be given a Riemannian metric, but they're not Riemannian manifolds until we pick one. We don't really talk about non-Riemannian manifolds, but "pseudo-Riemannian manifold" is close.

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    $\begingroup$ It sounds like the matter is really one of uniqueness, in the sense that there are many Riemannian manifolds, but there is only one Riemann integral? $\endgroup$
    – user856
    Feb 26, 2019 at 12:02
  • $\begingroup$ Actually, there is no book called "non-Reimannian Geometry." $\endgroup$
    – KCd
    Mar 7, 2019 at 2:50
  • $\begingroup$ @KCd fixed, thanks $\endgroup$
    – Mark S.
    Mar 7, 2019 at 12:37

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