On Local Null-Preserving Maps Between Pseudo-Riemannian Manifolds of Different Dimensions

We construct a local framework for maps between pseudo-Riemannian manifolds of different dimensions that preserve null directions. Let \( F: M_i \to M_{i+1} \); F is null-preserving if \( F_* v \) is null for every null \( v \in T_p M_i \). Deviations from an exact metric pullback are measured via a correction tensor T. This setup extends Liouville-type uniqueness results to the interdimensional case, providing a precise tool for local analysis of geometric embeddings.