geometry and topology

I was wondering what are the differences and relations:

between geometry and topology;

between differential geometry and differential topology;

between algebraic geometry and algebraic topology?

For example:

Are they studying different objects? Such as different mathematical structures/spaces?

if they study the same object, but study different aspects/properties of the same object?

...

Reading their wikipedia pages really confuses me.

Thanks and regards!

Solution 1:

As a supplement to Ryan's answer:

Differential geometry typically studies Riemannian metrics on manifolds, and properties of them. A typical differential geometry result is the sphere theorem, stating that if $M$ is a closed manifold equipped with a Riemannian metric for which the sectional curvatures lie in the half-open interval $(1/4, 1]\,\,$, then $M$ is a sphere. Note that the basic object is a manifold equipped with a Riemannian metric (a Riemannian manifold), and the curvature of the metric plays a key role in the statement of the theorem. These are typical features of problems/theorems in differential geometry. Note though that the conclusion of the theorem involves a statement about the topology of $M$; so there is certainly overlap between differential geometry and the concerns of topology. (One might say that the sphere theorem is a global result, using geometric hypotheses to draw topological conclusions. One can also have local results, in which topology plays no role in the hypothesis or conclusions: e.g. that a Riemanninan manifold with everywhere zero curvature is locally isometric to Euclidean space; one can also have global results that begin with topology and conclude with geometry: e.g. that any compact orientable surface of genus 2 or higher admits a Riemannian metric with constant curvature $-1$.)

Differential topology refers to results about manifolds that are more directly topological, and don't refer to metric structures. The generalized Poincare conjecture (that a closed manifold that is homotopic to a sphere is homeomorphic to one) is an example; another simpler example is Ehresmann's theorem, stating that a submersion between closed manifolds (or more generally, any proper submersion) is a fibre bundle.

Of course, these distinctions can be subtle, and may not always be well-defined, but a typical distinction between geometry and topology in general (and which is borne out in the preceding discussion) is that geometry studies metric properties of spaces, while topology studies questions which don't involve metric notions (it is the study of pure shape, if you like; the old name analysis situs also sheds some light on the meaning of topology).

One caveat is that, classically, Euclidean geometry branched not only into other geometries such as hyperbolic geometry (this branching being a precursor to Riemann's introduction of the general notions of Riemannian manifold and curvature), but gave rise to another branch known as projective geometry. In projective geometry, metric notions of distance and angle aren't studied (because they are not preserved by projective transformations), but notions such as being a straight line, or being a conic section, are. Algebraic geometry is the modern subject which developed out of projective geometry (among other sources; see this answer for a discussion of a quite different problem --- computing elliptic integrals --- which was another historical precursor to algebraic geometry). In algebraic geometry one studies varieties, which are solution sets to polynomial equations; thus in its elementary form it feels a lot like what is called analytic geomery in high-school, namely studying figures in the plane, or in space, cut out by equations in the coordinates.

Why is this geometry (as opposed to topology, say)? Because it turns out that when the functions one is using to cut out figures, or describe maps between figures, are restricted to be polynomial, the objects one obtains are quite rigid, in a way very similar to the way more traditional Euclidean geometry figures are rigid. So one has the sensation of doing geometry, rather than topology. (In topology, by contrast, things feel rather fluid, since one is allowed to deform objects in fairly extreme ways without changing their essential topological nature.) And in fact it turns out that there are deeper connections between algebraic and metric geometry: for example, for a compact orientable surface of genus at least 2, it turns out that the possible ways of realizing this surface as an algebraic variety over the complex numbers are in a natural bijection with the possible choices of a constant curvature -1 metric on the surface. A more recent example is given by the Calabi conjecture proved by Yau.

Solution 2:

Your question is quite vague. There are two parallel dynamics that go on when you learn mathematics. On one level, mathematics is extremely specific so as you learn one subject in detail it appears as if all you know is that subject and there appears to be no relation to any other subject. But once you get to understand something you start noticing patterns -- standard examples that appear in many different fields, but in different guises, standard constructions many of which fit into categorical or other natural frameworks that are beyond the specifics of one field. So to some extent there are broad unifying themes between subjects in mathematics. In that regard there's many connections between subjects labelled by names where you combine two of the words from the set {geometry(ic), topology, algebra(ic)}.

But at its most coarse, primitive level, there are some big differences. Algebraic geometry is about the study of algebraic varieties -- solutions to things like polynomial equations. Geometric topology is largely about the study of manifolds -- which are like varieties but with no singularities, i.e. homogeneous objects. Algebraic topology you could say is more about the study of homotopy-type or "holes in spaces". These are all inaccurate descriptions as in some sense subjects definitions are shaped by their histories. I'd say for example that Algebraic topology is more defined by the nature of the tools it employs. While geometric topology is more motivated by objects it wants to prove theorems about. That can seem like an artificial distinction, too, since isn't a "tool" an "object"? Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional phenomena being special is due to the existence of a big tool called the Whitney Trick, which allows one to readily convert certain problems in manifold theory into (sometimes quite complicated) algebraic problems. The thing is the Whitney trick fails in dimensions $4$ and lower. In that regard, geometric topology has some characteristics of a grumpy old man who is really set on figuring out something specific. And algebraic topology in some sense has more of the air of the person that follows the natural lay-of-the-land from some formal perspective.

For example, it's much more common in the geometric topology community to go to a talk that's an illustration of an idea, or a problem illustrated entirely by examples: generating tables of knots or a census of manifolds, or testing a hypothesis by computer experiement, etc.

But after all that is said, there are many links between these fields and others, so it frequently difficult to disambiguate them except in rather pat, artificial ways.

I'm not sure if that helps at all. But that's a generic 1st response.