In superstring theory, extra dimensions are conjectured. Then, the obvious observation that, macroscopically, we observe only three spacelike dimensions and one timelike dimension, leads to the requirement that, locally, the space-time of superstring theory admits a topological trivialitization of the form $\mathcal{M}\times \mathcal{K}$ where $\mathcal{M}$ is a four-dimensional Lorentzian manifold and $\mathcal{K}$ is small compact manifold (whose physical radius should be significantly smaller than $10^{-19}\ \text{m}$, the length scale currently being explored).
In most version of superstring theory $\mathcal{K}$ is taken as a Calabi-Yau Kähler manifold, who is a special case of Riemann manifold with zero Ricci curvature (Ricci flat). My question is:
What physical restrictions lead to the requierement of Ricci flatness of $\mathcal{K}$?
(this question is not a duplicate, I am asking for the reasons for zero Ricci curvature, not for the compactification itself).
