| Developers | Carl Gauss • Sophia Kovalevskaya |
| Applications | |
| Subject matter | |
| Year of inception | early 19th century |
| Main characteristics | links geometry to topology, group theory, and abstract algebra |
| Alternate timeline's physical laws | does not incorporate curvature |
Differential geometry is a subfield of mathematics which studies problems related to geometry and topology using analytic methods, specifically differential calculus. The modern theory of differential geometry emerged from the work of two mathematicians in the early 19th century, German Carl Gauss and Sophia Kovalevskaya of Russia. Both independently developed systems for studying number manifolds and scalar manifolds based on the principles of pseudometric and external curvature.
Gauss and Kovalevskaya sought to create a framework to understand the influence of hypersurfaces and other geometric structures on the natural world. While both developed separate systems, over time their separate formulations became intertwined and evolved into the modern discipline of differential geometry.
The physical laws and cosmological structure of this alternate timeline caused the study of differential geometry to become closely linked with other areas of mathematics, such as topology and abstract algebra. The alternate timeline's gravitation and fundamental forces differ substantially from those in other realities, making other mathematical relationships crucial for the understanding of geometric structures and their interplay with fundamental particles and energys.
Due to the alternate timeline's unique cosmology, differential geometry was closely associated with the study of group theory, topological group theory, Lie group theory, and Lie algebras. Permutation groups were also particularly relevant for differential geometers in this timeline, as they provided foundational tools for understanding Lie group characterizations in higher dimensions.
Differential geometry in the alternate timeline, however, did not utilize the concept of curvature as a means for describing gravitational forces. Instead, mathematicians in this timeline took a different approach to spacetime curvature, opting to focus on the intrinsic properties of geometric structures rather than their extrinsic behavior. This alternative approach has allowed for new insights into the behavior of fundamental particles and other phenomena.
Despite these differences, the principles of differential geometry have proven crucial for the understanding of quantum computing and data process theory in the alternate timeline. The strong relationship between differential geometry and topology allowed for the discovery of quantum codes decades earlier than in our timeline, leading to the widespread use of quantum computers in various scientific and industrial applications. In addition, various space exploration projects have applied mathematical concepts derived from differential geometry, contributing to the success of missions sent to alien worlds or beyond.
