3 Matching Annotations
 Dec 2021

www.quantamagazine.org www.quantamagazine.org

“[Floer] homology theory depends only on the topology of your manifold. [This] is Floer’s incredible insight,” said Agustin Moreno of the Institute for Advanced Study.
Floer homology theory depends only on the topology of the manifold.

 Jun 2021

math.stackexchange.com math.stackexchange.com

Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R
Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R where M is a smooth manifold is an object that produces, for any point p∈M and tangent vector v∈TpM, a number, the directional derivative, in a linear way. In other words, ==a differentiable function defines an element of the dual to the tangent space (the cotangent space) at each point of the manifold.==


en.wikipedia.org en.wikipedia.org

To put it succinctly, differential topology studies structures on manifolds that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that do have an interesting local (or sometimes even infinitesimal) structure.
Differential topology take a more global view and studies structures on manifolds that have no interesting local structure while differential geometry studies structures on manifolds that have interesting local structures.
