Why do I need to know topology to study differentional geometry?
I just try to understand differentional geometry, but I am not sure why topology is needed for it.
while I see that topology is an maybe interesting subject on itself , but ijust don't see what they have to do with eachother.
any reasons welcome :)
On
In a period of slightly over 100 years, we had a very successful theory (Gauss, inparticular) of surfaces in $\mathbb R^3,$ including curvature. This starts to split long before 1900: Riemann gave a description of a smooth metric, Beltrami in particular gave the first utterly convincing description of the non-Euclidean plane and space. This got the name of Poincare instead; meanwhile, Poincare developed the Analysis Situs, or algebraic topology. By the early 1900's Elie Cartan (and others, I imagine) had produced the definition of a smooth manifold, which brought the two sides back together.
Progress has not stopped, but that is where that topic lies. You can do a fair amount with submanifolds in Euclidean space; one undergraduate book, by John Thorpe, takes that viewpoint throughout. However, that places significant topological restrictions on what can be discussed.
Anyway, recommend emphasizing metric spaces as a special case of topology, and Riemannian manifolds early, and referring back to examples in Euclidean space as often as possible. The antidote to abstraction is plentiful concrete examples.
The importance of topology to differential geometry is a multifaceted one.
The language of point-set topology is simply too convenient for differential geometry. The major theorems of topology simplify statements and proofs of important theorems in differential geometry. The intermediate value theorem (continuous functions send connected sets to connected sets) is one example of a point-set topology statement that has enormous consequences in differential geometry.
Many constructs in differential geometry are easiest to define with the help of topology. The basic objects, differentiable manifolds, are themselves examples of second-countable Hausdorff spaces, and fiber bundles are topological spaces that are at the heart of the general theory of manifolds. One at least needs to understand the idea of a continuous map of topological spaces to define a section of a fiber bundle, which is key to defining vector fields, connections, and consequently all of the important stuff like curvature.
The manifolds of greatest interest often should be considered as more than just manifolds. Many of them are topological groups; Lie groups are examples of manifolds with a (more than) topological group structure, which lends additional structure that you may wish to analyze. It is common to put topological restrictions on manifolds: the common ones are compactness and connectedness. Completeness is another (and for Riemannian manifolds completely characterized by the Hopf-Rinow theorem).
One of the major goals for differential geometry is to show how localized information about a manifold translates into global, i.e. topological implications for the manifold. The Gauss-Bonnet theorem, which relates the Gauss curvature of a surface (a local, analytic property) to the Euler characteristic of a manifold (a global, topological property that completely classifies surfaces topologically), is one example. Some of the most powerful theorems in differential geometry (Atiyah-Singer, sphere theorem) are of this form. They allow you to go two ways: if you understand the topology of a manifold, you can greatly restrict its geometry (the curvature usually), while if you know the curvature at a few select points/regions of a manifold then you may be able to say something about its topology.
There are other ways in which the topology of a manifold is linked to its differential geometric properties. I don't want to make a long list, so I'll just mention one that sticks out to me: one can define integration on manifolds via differential forms, which relates the cotangent bundle of a manifold to its de Rhan (and hence singular) cohomology.