Differential Topology
Differential topology is a branch of mathematics that uses tools from calculus to study the properties and structures of differentiable manifolds, which are spaces that can be locally modeled by open subsets of Euclidean space. This field is concerned with the geometric and topological properties of these manifolds that are preserved under smooth deformations. Differential topology has applications in fields such as physics, engineering, and computer graphics. As a subfield of topology, differential topology focuses on those aspects of manifolds that are preserved under smooth maps, while other branches of topology study more general classes of continuous transformations. Within the hierarchy of science, mathematics, and topology, differential topology is a specialized area that uses calculus and geometry to study the properties of differentiable manifolds, providing a deeper understanding of their structure and behavior.