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

Solution 1:

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.