Note on Almost Product Manifolds and their Tangent Bundles

Abstract

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.