Suppose that $X$ and $\check{X}$ are Calabi-Yau manifolds mirror to each other. Then

• (i)

  both $X$ and $\check{X}$ admit special Lagrangian torus fibrations with sections $\mu:X\to B$ and $\check{\mu}:\check{X}\to B$ over the same base:

• (ii)

  the fibrations $\mu:X\to B$ and $\check{\mu}:\check{X}\to B$ are fiberwise dual to each other in the sense that if the fibers $\mu^{-1}(b)\subset X$ and $\check{\mu}^{-1}(b)\subset\check{X}$ over $b\in B$ are nonsingular, then they are dual tori; and

• (iii)

  there exist fiberwise Fourier(-Mukai)–type transforms which are responsible for the interchange between the symplectic-geometric (resp. complex-geometric) data on $X$ and the complex-geometric (resp. symplectic-geometric) data on $\check{X}$.