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I am a grad student. I am writing an article on geometry and relativity theory and trying to start with discussing basic ideas of topology. In my article I tried very hard to motivate the idea of topology and make it natural as much as possible. My plan in my article is to move then from topology to geometry. But surprisingly, I did not find in any book I looked at so far a clear distinction between topology and geometry other than giving a topological space a distance structure(or a metric structure). The idea of imposing a metric structure on a topological space perfectly fits in the context of geometry for physicists. But, how would a mathematician make the a distinction between topology and geometry?, or in other words, how can we define a geometry on a topological space?. Any suggestions for references or books would be really appreciated along with your comments.

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Roughly speaking, you could say that geometry is concerned with "rigid" properties whereas topology deals with "fluid" ones. – Adeel Mar 6 '13 at 10:00
I second what Adeel says. You may also want to read about the Erlangen program, which was the first attempt to answer what geometry is after the proliferation of non-Euclidean geometries in the nineteenth century. You'll also note that the notion of metric is not central. – Raskolnikov Mar 6 '13 at 10:02
It's not exactly the same question, but you might also be interested to look at – Matthew Pressland Mar 6 '13 at 10:26
in addition to @Adeel, I would add things like Minkowskian structures, symplectic structures and Poisson structures which should be considered under "geometry". – magguu Mar 6 '13 at 16:20
This is a grand question. You made a big mistake in accepting the first answer immediately. – Christian Blatter Mar 6 '13 at 16:33
up vote 1 down vote accepted

I like to think of the distinction between topology and geometry using symmetry, in the spirit of the Erlangen program. The idea is that you give an object more structure by being more strict about what you consider its symmetries to be. Suppose, for example, you're asking a question about the $2$-sphere. If the answer to your question depends only on the homotopy or homeomorphism type of the sphere, then you're doing topology. If the answer is invariant under the diffeomorphism group of the sphere (very large, but smaller than the homeomorphism group), then you're doing differential topology. If the answer is only invariant under the isometry group - the finite dimensional Lie group $O(3)$ - then you're doing Riemannian geometry. Finally, maybe your question is about how the sphere interacts with the integer lattice in $\mathbb{R}^3$; then you're doing algebra, and your symmetry group is likely to be finite.

This way of thinking does not necessarily provide a universal distinction between geometry and topology; some crappy metric spaces have infinite dimensional isometry groups. But it is backed by theorems: for example, the isometry group of any Riemannian manifold is a finite dimensional Lie group while the full diffeomorphism group is infinite dimensional.

This is also very much in line with how physicists think. The standard model is to a large extent about characterizing and classifying tiny particles according to their symmetries. Gauge theory is about relating the symmetries of a physical system to its fundamental laws. In fact, the "general" in general relativity actually has to do with symmetry: Einstein wanted wanted his field equations to be "generally covariant", i.e. they should look the same under any smooth change of coordinates, i.e. they should be diffeomorphism invariant. In fact this is not possible for general relativity, and this is why Riemannian geometry and not differential topology is the right language to describe it.

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Topology is part of the general study of geometry. Roughly, topology is the study of the fine structure of a space, metric geometry is the study of properties of a space in terms of distances, and coarse geometry is the study of the large scale structure of a space. A metric space gives rise to both a topology and a coarse structure. Passing from a metric space to the topology induced by it loses a lot of information, and in a sense only retains the fine scale structure. Similarly, passing from a metric space to the coarse structure induced by it looses a lot of information and retains the large scale structure.

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To me the relevant questions defining any branch of geometry or topology are "what are the shapes" and "what are the rules for determining when two shapes are the same?" I don't know a way of making this rigorous, but a branch of math that studies shapes is some sort of "geometry" if it's difficult for two shapes to be the same, and it's some sort of "topology" if it's easy.

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Maybe the best example of this: if we demand that our topological spaces be (smooth) real manifolds, and we want to study only smooth maps between them, we're definitely still studying topology, possibly "differential topology." Now take what appears to be the same definition (formally) except replacing real by "complex" and suddenly we're studying "complex geometry." The paucity of holomorphic maps (I stole this phrase from somewhere, but I don't know where!) makes life geometric where before it was merely topological. – user29743 Mar 6 '13 at 18:36

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