I found it a problem when searching references for Riemannnian Geometry, as I wrongly typed as Riemann Geometry just like Riemann Surface.
So are there any criteria to decide if we should use Riemann or Riemannian?
For example, Riemann in these phrases:
Riemann surface, Riemann curvature tensor, Riemann-integrable, Riemann sum, Riemann sphere, Riemann-Roch theorem, Riemann Hypothesis...
For example, Riemannian in these phrases:
Riemannian geometry, Riemannian manifold, Riemannian measure, Riemannian metric...
Especially, why it's Riemann surface but Riemannian manifold?
And why it's Riemann-Roch theorem, Riemann Hypothesis but Fermat's Last theorem ?
In addition, what about Abel vs Abelian?
(This is a little different, since Abelian = commutative in most cases)
And Gauss vs Gaussian?
Gauss curvature, Gauss sum, Gauss Theorem... (I think it's named after Gauss)
Gaussian measure, Gaussian variable, Gaussian distribution, Gaussian integer, Gaussian process...
$1.$ 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 some 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 typo about "Riemann" in this website. Till now, there're $17$ "Reiman", $268$ "Reimann" and $22$ "Reimannian" in MSE.