Differential geometry is foundational in general relativity because it provides the mathematical language to explain the curvature of space-time, which is at the core of gravity in the framework of general relativity. Traditional methods viewed gravity as a force in the context of a flat background and did not explain several observed phenomena or reported natures of physics (Wald, 1984). General relativity explains how gravity pulls objects towards the earth's center because it delineates gravity as a consequence of a planet's mass.

Differential geometry provides different tools like connections, Riemannian curvature tensors, geodesics that are essential for systematically explaining curvature. The Riemann curvature tensors, for instance, reinforce the understanding of the manner in which space-time bends around immense objects (Misner et al., 1973). Also, advances in differential geometry, including the understanding of geometric flows and global analysis, help physicists explain the global structure of solutions to the field equations proposed by Einstein (Chow et al., 2007).

These advances inform the explanation of phenomenon not clear in other disciplines, especially in string theory and brane-world scenarios (Nakahara, 2003). Therefore, differential geometrics demonstrates that the law of gravity is a fundamental part of our success in understanding physics as a product of the universe's fundamental framework rather than a spontaneous rule.

References

Chow, B., Chu, S.–C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Kraft, D., & Knopf, D. (2007). The Ricci Flow: Techniques and Applications. Department of Mathematics, John Hopkins University.

Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W.H. Freeman and Co.

Nakahara, M. (2003). Geometry, Topology and Physics. Institute of Physical Publishing. Wald, R. M. (1984). General relativity. University of Chicago Press.