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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.
We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or $CH_3$ for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, $CH_2$ manifolds that are not homogeneous.
We consider pointed Lorentzian manifolds and construct canonical foliations by constant mean curvature (CMC) hypersurfaces. Our result assumes a uniform bound on the local sup-norm of the curvature of the manifold and on its local injectivity radius,
We consider three-dimensional Lorentzian metrics that locally admit four independent Killing vectors. Their classification is summarized, and conditions for characterizing them are found. These consist of algebraic classification of the traceless Ric
In this paper, we investigate conformal Killings vectors (CKVs) admitted by some plane symmetric spacetimes. Ten conformal Killings equations and their general forms of CKVs are derived along with their conformal factor. The existence of conformal Ki
We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike sing