Geometrical Theory on Combinatorial Manifolds

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For an integer m≥ 1, a combinatorial manifold M is defined to be a geometrical object M such that for ∀ p∈M, there is a local chart (Uₚ,ϕₚ) enable ϕₚ:Uₚ→ Bⁿᵢ₁ Bⁿᵢ₂... Bⁿᵢₛ₍ₚ₎ with Bⁿᵢ₁ Bⁿᵢ₂... Bⁿᵢₛ₍ₚ₎=, 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.