# $\mathcal{I}$-Preserving Diffeomorphisms of Lorentzian Manifolds

## TL;DR

This paper investigates the existence and structure of $ ext{I}$-preserving diffeomorphisms in Lorentzian manifolds, linking them to nil-Killing vector fields and exploring their role in degenerate Kundt spacetimes with constant scalar invariants.

## Contribution

It characterizes conditions under which $ ext{I}$-preserving diffeomorphisms exist and relates them to nil-Killing vector fields in Lorentzian manifolds, especially in degenerate Kundt spacetimes.

## Key findings

- $ ext{I}$-preserving diffeomorphisms exist only in $ ext{I}$-degenerate spacetimes.
- Nil-Killing vector fields form Lie algebras, possibly infinite, but not all generate $ ext{I}$-preserving diffeomorphisms.
- Finite transitive Lie algebra of nil-Killing vector fields exists in Kundt spacetimes with constant scalar polynomial invariants.

## Abstract

We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be related to nil-Killing vector fields, for which the associated Lie derivative of the metric yields a nilpotent rank two tensor. We show that the set of nil-Killing vector fields contains Lie algebras, although the Lie algebras may be infinite and can contain elements which are not $\mathcal{I}$-preserving diffeomorphisms. We then study the curvature structure of a general Lorenztian manifold, or spacetime, to show that $\mathcal{I}$-preserving diffeomorphism will only exists for the $\mathcal{I}$-degenerate spacetimes and to determine when the $\mathcal{I}$-preserving diffeomorphisms are generated by nil-Killing vector fields. We identify necessary and sufficient conditions for the degenerate Kundt spacetimes to admit an additional $\mathcal{I}$-preserving diffeomorphism and conclude with an application to the class of Kundt spacetimes with constant scalar polynomial curvature invariants to show that a finite transitive Lie algebra of nil-Killing vector fields always exists for these spacetimes.