With commuting Killing vectors, the lapse and shift of one Killing vector are constants along the other

TL;DR

This paper proves that in a manifold with two commuting Killing vectors, the lapse and shift of one vector are constant along the orbits of the other, revealing conserved quantities related to symmetries.

Contribution

It establishes that certain geometric quantities remain constant along specific Killing vector flows in manifolds with commuting symmetries.

Findings

Lapse and shift of one Killing vector are constant along the other.

Six dot products involving Killing vectors and the normal are conserved along the surface Killing vector.

Provides a geometric understanding of conserved quantities in symmetric manifolds.

Abstract

Given an d-dimensional manifold with two commuting Killing vectors, together with an d - 1 dimensional submanifold in which one of the Killing vectors lies, then the lapse and shift of the second Killing vector, relative to this slice, remain constant along the orbits of the `surface' Killing vector. Alternatively, the six dot products that can be formed from the three vectors, the two Killing vectors and the normal to the submanifold, are all constants along the `surface' Killing vector.