A Pseudo-Riemannian Generalization of Euclidean Subspace Hierarchies

We propose a unifying framework that generalizes classical Euclidean subspace hierarchies into a pseudo-Riemannian landscape by introducing a signature-based stratification of manifolds. For a total dimension D = m+n, we systematically construct the space of submanifolds Mm,n with m spacelike and n timelike dimensions. These give rise to a structured family of pseudo-Grassmannians Gr(m,n)(M,N), where signature plays a central geometric and physical role. We extend key constructions— such as Pl¨ucker embeddings, local charts, homogeneous space representations, volume forms, and cohomological invariants—to these indefinite-signature settings. Furthermore, we explore implications for symmetry breaking, field theory on signature-changing manifolds, and brane-world cosmologies involving M(1,3), M(2,2), and M(3,1) universes. Applications to twistor theory, supersymmetry, and quantum gravity foams with signature fluctuations are included. The resulting geometric machinery provides new tools for modelling transitions between classical, subtle, and metaphysical layers of spacetime.