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

TL;DR

This paper introduces modified Gauss-Codazzi-Ricci equations tailored for Kaluza-Klein reduction and the Cauchy problem, highlighting the role of an antisymmetric extrinsic curvature tensor component.

Contribution

It presents a novel variation of classical equations incorporating antisymmetric parts, advancing the mathematical framework for higher-dimensional theories and general relativity.

Findings

The extrinsic curvature tensor includes both symmetric and antisymmetric parts.

In certain limits, the extrinsic curvature reduces to a purely antisymmetric tensor.

The variations are suitable for applications in string theory and gravitational problems.

Abstract

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.