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?

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.
 A: 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.
