The mirror map and other natural tangent tensors

We review the intrinsic geometry of the tangent bundle of a differentiable manifold $M$, aside from any non-natural structures. We recall the properties of the mirror map $B\in\mathrm{End}(TTM)$, known also as the canonical endomorphism or the almost-tangent structure, solve some cohomological questions and raise other induced by the $\mathrm{d}_B$ operator.