Differential Geometry
Differential geometry is a branch of mathematics that studies the properties and structures of curves, surfaces, and more generally manifolds in a space that can be Euclidean or non-Euclidean. It involves the study of calculus on these geometric objects, leading to concepts such as differentials, tangent spaces, curvature, and geodesics. Differential geometry has deep connections with physics, particularly in the study of general relativity, where the curvature of spacetime is used to describe gravity. In this hierarchy, differential geometry fits under concrete mathematics because it deals with precise mathematical structures that can be defined using equations and formulas. It also falls under mathematics and science due to its foundational role in understanding geometric shapes and their properties as well as its applications in various scientific fields such as physics and engineering.