An axiom is said smarandachely denied if in the same space the axiom behaves differently (i.e., validated and invalided; or only invalidated but in at least two distinct ways). Therefore, we say that an axiom is partially negated, or there is a degree of negation of an axiom.
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.
The most important contribution of Smarandache geometries was the introduction of the degree of negation of an axiom (and more general the degree of negation of a theorem, lemma, scientific or humanistic proposition) which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood) or more general like the negation in neutrosophic logic (with a degree of truth, a degree of falsehood, and a degree of neutrality (neither true nor false, but unknown, ambiguous, indeterminate) [not only Enclid’s geometrical axioms, but any scientific or humanistic proposition in any field] or partial negation of an axiom (and, in general, partial negation of a scientific or humanistic proposition in any field).
These geometries connect many geometrical spaces with different structures into a heterogeneous multispace with multistructure.
In general, a rule R ∈ R in a system (Σ; R ) is said to be Smarandachely denied if it behaves in at least two different ways within the same set Σ, i.e. validated and invalided, or only invalided but in multiple distinct ways.
A Smarandache system (Σ;R ) is a system which has at least one Smarandachely denied rule inR .
In particular, a Smarandache geometry is such a geometry in which there is at least one Smarandachely denied rule, and a Smarandache manifold (M;A) is an n-dimensional manifold M that supports a Smarandache geometry.
In a Smarandache geometry, the points, lines, planes, spaces, triangles, ... are respectively called s-points, s-lines, s-planes, s-spaces, s-triangles, ... in order to distinguish them from those in classical geometry.
Howard Iseri constructed the Smarandache 2-manifolds by using equilateral triangular disks on Euclidean plane R2. Such manifold came true by paper models in R3 for elliptic, Euclidean and hyperbolic cases. It should be noted that a more general Smarandache n-manifold, i.e. combinatorial manifold and a differential theory on such manifold were constructed by Linfan Mao.
Nearly all geometries, such as pseudo-manifold geometries, Finsler geometry, combinatorial Finsler geometries, Riemann geometry, combinatorial Riemannian geometries, Weyl geometry, Kähler geometry are are particular cases of Smarandache geometries.
[Dr. Linfan Mao, Chinese Academy of Sciences, Beijing, P. R. China, 2005-2011]
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