Kaluza-Klein dimensional reduction and Gauss-Codazzi-Ricci equations

TL;DR

This paper derives the Gauss-Codazzi-Ricci equations for Kaluza-Klein dimensional reduction using a geometric approach, clarifying the relationship between extrinsic curvature and gauge fields, and exploring symmetries and instantons.

Contribution

It introduces a geometric derivation of the Gauss-Codazzi-Ricci equations in Kaluza-Klein theory, emphasizing the geometric meaning and analyzing gauge symmetries and instantons.

Findings

Extrinsic curvature proportional to gauge field strength when lower-dimensional metric is independent of extra dimensions.

Discussion of SO(n) and SU(n) symmetries in the internal space.

Analysis of Kaluza-Klein instantons.

Abstract

In this paper we imitate the traditional method which is used customarily in the General Relativity and some mathematical literatures to derive the Gauss-Codazzi-Ricci equations for dimensional reduction. It would be more distinct concerning geometric meaning than the vielbein method. Especially, if the lower dimensional metric is independent of reduced dimensions the counterpart of the symmetric extrinsic curvature is proportional to the antisymmetric Kaluza-Klein gauge field strength. For isometry group of internal space, the SO(n) symmetry and SU(n) symmetry are discussed. And the Kaluza-Klein instanton is also enquired.