On the possibility of a Erbacher-type reduction in the geometric theory of spacetimes

Although Erbacher’s reduction Theorem on Riemannian isometric immersions in the space forms is known to hold in the realm of semi-Riemannian geometry, the hypotheses cannot be translated to the non-relativistic context of Galilean geometry. In this paper, we prove a Erbacher-type reduction result in both Lorentzian and Galilean frameworks: the worldline of any Unchanged Direction observer in a spacetime with certain infinitesimal symmetries can be embedded in a 3-dimensional totally geodesic submanifold with Galilean or Lorentzian structure. • An UD observer is a future-directed timelike curve with parallel unit normal vector. • A spacetime endowed with a torse-forming field admits a twisted decomposition. • CSC restspaces ensure UD observers are in a totally geodesic 3-manifold.