Killing Operator for the Kerr Metric

When D: E → F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J_q(E) → F=F₀ that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D₁: F₀ → F₁. When D is involutive, that is when the corresponding system R_q = ker (Φ) is involutive, this procedure provides successive first order involutive operators D₁, ..., D_n. Though D₁ ο D = 0 implies ad (D) ο ad(D₁) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D₁) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R_q is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the