O ct 2 00 6 Pseudo - Manifold Geometries with Applications ̧

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969), i.e., an axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways and a Smarandache n-manifold is a n-manifold that support a Smarandache geometry. Iseri provided a construction for Smarandache 2-manifolds by equilateral triangular disks on a plane and a more general way for Smarandache 2-manifolds on surfaces, called map geome-tries was presented by the author in [9] − [10] and [12]. However, few observations for cases of n ≥ 3 are found on the journals. As a kind of Smarandache geometries, a general way for constructing dimensional n pseudo-manifolds are presented for any integer n ≥ 2 in this paper. Connection and principal fiber bundles are also defined on these manifolds. Following these constructions, nearly all existent geometries, such as those of Euclid geometry, Lobachevshy-Various geometries are encountered in update mathematics, such as those of Euclid geometry, Lobachevshy-Bolyai geometry, Riemann geometry, Weyl geometry, Kähler geometry and Finsler geometry, ..., etc.. As a branch of geometry, each of them has been a kind of spacetimes in physics once and contributes successively to increase human's cognitive ability on the natural world. Motivated by a combinatorial notion for sciences: combining different fields into a unifying field, Smarandache introduced neutrosophy and neutrosophic logic in references [14]−[15] and Smarandache geome-tries in [16]. An axiom is said to be Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways. A Smarandache geometry is a geometry which has at least one Smarandachely