Variations of Gauss-Codazzi-Ricci Equations in Kaluza-Klein Reduction (String Theory) and Cauchy Problem (General Relativity)

We find a kind of variations of Gauss-Codazzi-Ricci equations suitable for Kaluza-Klein reduction and Cauchy problem. Especially the counterpart of extrinsic curvature tensor has antisymmetric part as well as symmetric one. If the dependence of metric tensor on reduced dimensions is negligible it becomes a pure antisymmetric tensor.