In their proposed compactification of superstring [4], Candelas, Horowitz, Strominger and Witten took the matric product of a maximal symmetric four dimensional spacetime M with a six dimensional Calabi-Yau vacua X as the ten dimensional spacetime; they identified the Yang-Mills connection with the SU(3) connection of the Calabi-Yau metric and set the dilaton to be a constant. To make this theo...

We find geometric conditions on a four-dimensional Hermitian manifold endowed with metric connection totally skew-symmetric torsion under which the complex structure is harmonic map from into its twistor space considered natural family of Riemannian metrics defined by means and given base manifold.