Classical and Differential Topological Approaches to Gravitational Fields and Cosmic Singularities

This paper explores the relationships between differential topology and general relativity, examining the classical framework's applicability in modelling gravitational fields and the limits encountered with topological defects such as cosmic strings. By illustrating smooth metric structures, holonomy, and geodesic completeness, we reveal the foundational role of differential topology in describing spacetime curvature without quantum corrections. The Einstein field equations and global topological invariants are discussed in contexts where continuous manifolds hold, while highlighting how phenomena like cosmic strings challenge these assumptions. We propose that such topological defects introduce singular behaviours where traditional differential topology reaches its boundaries, potentially necessitating quantum gravitational models for a complete description. Visualizations of Schwarzschild curvature, holonomy in spherical surfaces, and geodesic paths provide insights into gravitational field variability and topological constraints in classical models, underscoring the mathematical and physical principles at play.This paper explores the relationships between differential topology and general relativity, examining the classical framework's applicability in modelling gravitational fields and the limits encountered with topological defects such as cosmic strings. By illustrating smooth metric structures, holonomy, and geodesic completeness, we reveal the foundational role of differential topology in describing spacetime curvature without quantum corrections. The Einstein field equations and global topological invariants are discussed in contexts where continuous manifolds hold, while highlighting how phenomena like cosmic strings challenge these assumptions. We propose that such topological defects introduce singular behaviours where traditional differential topology reaches its boundaries, potentially necessitating quantum gravitational models for a complete description. Visualizations of Schwarzschild curvature, holonomy in spherical surfaces, and geodesic paths provide insights into gravitational field variability and topological constraints in classical models, underscoring the mathematical and physical principles at play.

General Relativity; Differential Topology; Gravitational Fields; Cosmic Strings; Geodesics Holonomy; Schwarzschild Solution; Topological Defects; Einstein Field Equations; Curvature