2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131695The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,ϕ,Λ), with the calibration 3-form ϕand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define various different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can define mirror dual G_2 manifolds inside of a Spin(7) manifold (N^8, Ψ). In case N^8 admits an oriented 3-plane field, by iterating this process we obtain Calabi-Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G_2 (or Spin(7)) manifold.Revised version, remarks added, 20 pages, 3 figuresDifferential GeometryHigh Energy Physics - TheoryGeometric Topologytext