Can we define gravity on Calabi-Yau manifolds? I have read about applying Hermitian geometry in general relativity in deriving holomorphic gravity. But if we take it some steps further i.e. allowing Kähler manifolds with the Ricci flatness condition, can we consider general relativistic gravitational force on Calabi-Yau manifolds?
 A: The physical point of Calabi-Yau manifolds being Ricci flat is that a string wrapped on it will not spread. The Hamilton equation for Ricci flows
$$
\frac{dg_{ij}}{dt}~+~2R_{ij}~=~0.
$$
If there is a nonzero Ricci curvature then the manifold will change its geometry with time. This will mean that a string wrapped on this space will have its world sheet area change, and thus is action according to the Goto theorem. In general a string will spread out over the space. 
If general relativity contributed to the Calabi-Yau manifold curvature, say if there is something as an inhomegenous term to the Hamilton equation, much the same would happen. This might be problematic. However, gravitational wave data on GRB 150101B indicates a very low bound on gravitational waves leaking into so called extra large dimensions. It then appears that data does not support the idea of gravitational waves leaking into other dimensions, say a Calabi-Yau manifolds, and causing some of the theoretical mischief I point out here.
