Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective mathematics Relating to the branch of
geometry inspired by Bernhard Riemann.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

Geometry, defining it as a duality between a general manifold and a socalled Riemannian metric.

“Habilitationsschrift” (1868); and as an immediate ap  plication of it he outlined the socalled Riemannian

Life as it exists at any point in physicalspacetime, is a finite domain, but, by virtue of what were usefully identified as Riemannian evolution, is not a bounded domain.

This qualification is, that all competent modern science is Riemannian in that coincidental sense of the use of the term "Riemannian" by both Einstein's treatment of the subject of Kepler's astronomy, and in the related case of Academician V.I. Vernadsky's defining of the physical chemistry of the Noösphere.

Some informal background: a Riemannian manifold is a differentiable manifold (where the tangent space at each point has an inner product) with a positivedefinite metric tensor, d (x, y) ≥ 0.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

A familiar Riemannian manifold is a Euclidean manifold (where one has to add a smoothly varying inner product on the tangent space of the standard Euclidean space), with the familiar Euclidean (distance) metric (our 3space, for example).
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

However, the General Theory of Relativity explains the concept of gravitation by using Riemannian geometry.
Matthew Yglesias » Cato’s David Boaz Joins George Will in Peddling Bogus “Global Cooling” Stories

The Lorentzian manifold is a pseudoRiemannian manifold, the generalization of the Riemannian manifold, such that the metric tensor need not be positivedefinite.
Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element

“However, the General Theory of Relativity explains the concept of gravitation by using Riemannian geometry”
Matthew Yglesias » Cato’s David Boaz Joins George Will in Peddling Bogus “Global Cooling” Stories

Riemannian writes: Bryan, have you noticed any more Ivory Tower academics more partial to blogging as a result of the increasingly prominent stature of the GMU economics department on the blogosphere, especially with the attention yours 'and Tyler's books are receiving?
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