An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space (i.e., validated and invalided, or only invalidated but in multiple ways).

A SMARANDACHE GEOMETRY is a geometry which has at least one Smarandachely denied axiom (1969).

Thus, as a particular case, Euclidean, Lobachevsky-Bolyai-Gauss, and

Riemannian geometries may be united altogether, in the same

space, by some Smarandache geometries.

These last geometries can be partially Euclidean and partially

Non-Euclidean.

It seems that Smarandache Geometries are connected with the

Theory of Relativity (because they include the Riemannian

geometry in a subspace) and with the Parallel

Universes.

Papers, notes, comments are welcome and

will be published in a collection book.

See: http://www.gallup.unm.edu/~smarandache/geometries.htm