# Criterion for using "Riemann" or "Riemannian" in terminology

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?

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.

• 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. Feb 26, 2019 at 10:39
• Has showed by the many links in yoyr post, natural language is not always "logical"; habits and conventions rule. Feb 26, 2019 at 10:40
• 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. Feb 26, 2019 at 10:50
• And of course Cauchy-Reimann as well, reimann-integrable, Reimann Mapping Theorem, Lower Reimann Integral etc. Feb 26, 2019 at 10:53
• French isn't consistent either, but it seems to be consistent with English : we have "surface de Riemann" but "variété Riemanienne" for instance Feb 26, 2019 at 11:48

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.

• 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?
– user856
Feb 26, 2019 at 12:02
• Actually, there is no book called "non-Reimannian Geometry."
– KCd
Mar 7, 2019 at 2:50
• @KCd fixed, thanks Mar 7, 2019 at 12:37