Differential topology versus differential geometry(Wiki)

Differential topology and differential geometry are first characterized by their similarity. They both study primarily the properties of differentiable manifolds, sometimes with a variety of structures imposed on them.

One major difference lies in the nature of the problems that each subject tries to address. In one view,^[3] differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global. Consider the example of a coffee cup and a donut (see this example). From the point of view of differential topology, the donut and the coffee cup are the same (in a sense). This is an inherently global view, though, because there is no way for the differential topologist to tell whether the two objects are the same (in this sense) by looking at just a tiny (local) piece of either of them. He or she must have access to each entire (global) object.

From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut. This is also a global way of thinking about the problem. But an important distinction is that the geometer does not need the entire object to decide this. By looking, for instance, at just a tiny piece of the handle, he can decide that the coffee cup is different from the donut because the handle is thinner (or more curved) than any piece of the donut.

Đa tạp vi phân và hình học vi phân

Loại bỏ các tính chất riêng biệt của các đối tượng hình học được nghiến cứu trong hình học vi phân, tổng quát hóa các tính chất chung nhất của chúng, người ta đi đến khái niệm đa tạp vi phân chứa các khái niệm về các đường, các mặt, họ các đường, các mặt trong không gian Euclide và phi Euclide. Như vậy, các đa tạp vi phân chính là các tượng tổng quát của hình học vi phân.