Mathematical Foundations of Constrained SO(3,3) Spacetimes: Part I. Isometric Embeddings and Topological Surgery

Abstract

Multitemporal manifolds (e.g., SO(3,3)) are conventionally plagued by causality violations, pathological Cauchy problems, and the inevitable formation of closed timelike curves (CTCs). In this paper, we establish a rigorous differential geometric framework that resolves these pathologies without invoking spatial compactification, moduli stabilization, or dynamic field equations. By modeling physical world-volumes as developable submanifolds subject to the Wunderlich isometric embedding limit (η ≤ 1/√3), we mathematically prove that transverse temporal degrees of freedom are strictly bounded. We show that this extrinsic curvature constraint acts as a rigid kinematic barrier, geometrically enforcing global hyperbolicity and strictly prohibiting the formation of CTCs within the effective projection. Furthermore, we demonstrate that attempting to exceed this geometric bound results in the breakdown of diffeomorphism, necessitating topological surgery that yields non-orientable, soliton-like boundary conditions (Möbius loops). This work provides a complete mathematical foundation for dimensionally constrained spacetimes purely through extrinsic differential geometry.