Geometrical Theory on Combinatorial Manifolds

Abstract

For an integer m ≥ 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for (...), there is a local chart (see paper) where Bnij is an nij -ball for integers 1 ≤ j ≤ s(p) ≤ m. Topological and differential structures such as those of d-pathwise connected, homotopy classes, fundamental d-groups in topology and tangent vector fields, tensor fields, connections, Minkowski norms in differential geometry on these finitely combinatorial manifolds are introduced. Some classical results are generalized to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed and geometrical inclusions in Smarandache geometries for various geometries are also presented by the geometrical theory on finitely combinatorial manifolds in this paper.