Geometry - Topology; What is the difference?

[member 11137]

Geometry - Topology; "What is the difference?"

It is certainly important for a good understanding of a lot of modern problems. So I think it could be important to explain clearly the difference(s) between these two notions. Can you help me?

 

[HallsofIvy]

What do you mean by "geometry"? "Topology" has a pretty standard definition- there is even a specific definition for "a topology" but "geometry" is a pretty general term. If forced to give a specific definition, I would say that "geometry" is the study of general "figures" while "topology" is a subset of "geometry".

 

[tom.stoer]

Think about a large and a small circle with radius R and r, respectively. They are both circles, so they have the same topology, but they have different size (radius R and r; length 2πR and 2πr) so their geometry differs.

Now think about a circle of radius R and a line segment of length 2πR; they are topologically different (you can walk around the circle whereas the line segment has two endpoints). They have one geometrical property in common, namely the length L=2πR, but due the different topology the meaning of "length" is different for the circle and for the line segment.

 

[Demystifier]

Without intention to be rigorous or precise, topology can be viewed as a part of geometry studying properties of geometrical objects which do not change under continuous deformations of these objects.

 

[member 11137]

Geometry seems (in general) to be related to the concept of "distance" whilst topology seems to be related to the notion of "form". More precisely: geometry is related to the measurement of distances of or on objects whilst topology is interested in the description of the forms of these objects. Starting from this distinction, I remark that most of courses devoted to topology start with the definition of what a distance is. This is implicitly revealing that a distance is a good tool to measure the characteristics of a form and that variations of a distance are a good tool to measure the deformations of the same form.

I ignore if that representation is correct but it is actually mine (for HallsofIvy). I really appreciate the pictural help (for tom.soer) and the qualitative precision (for Demystifier).

Now the difficulty in mathematics comes from the fact that objects are sometimes abstract objects (e.g.: the phase space) and don't exactly correspond to real one (e.g.: a ball, a table...), i.e. to objects that we can effectively carry with our hands.

Thank you

 

[mathman]

This seems to be a mathematics question. I think it is in the wrong forum.

 

[micromass]

Geometry seems (in general) to be related to the concept of "distance" whilst topology seems to be related to the notion of "form". More precisely: geometry is related to the measurement of distances of or on objects whilst topology is interested in the description of the forms of these objects. Starting from this distinction, I remark that most of courses devoted to topology start with the definition of what a distance is. This is implicitly revealing that a distance is a good tool to measure the characteristics of a form and that variations of a distance are a good tool to measure the deformations of the same form.

Hmm, that's a good insight but I don't find it too accurate. Not all geometries are concerned about distance. For example, projective geometry doesn't care about distance at all. What projective geometry is, is actually projecting a 3D world on a 2D plane (for example, a painter want to accurately represent the 3D world on his painting). But this projection doesn't care about distance at all: indeed, things which are far away might get projected close together. In the same way, they don't care about angles. But still it is quite remarkable that the basic shape of an object is still recognizable from the projection. Why exactly that is and what properties of the shape are being preserved is the study of projective geometry.

Klein's Erlangen program actually says that a geometry is being defined by a certain set of transformation. These transformations determine what is being kept constant and what is being varied.
For example, if you take as transformations the distance preserving maps, then you get a geometry where distance is an important invariant. If you would rather take the projective transformations, then you get projective geometry, etc.

Klein's program is a bit outdated right now, but I feel that it still has some value.

A good book that you might enjoy reading is "Geometry and Topology" by Reid and Szendroi.

 

[Tac-Tics]

The -metry suffix of geometry means measurement. Geometry is the study of shapes and their quantitative measurements. What is the angle between these two lines? How many sides does this thing have? What is the shortest path following the surface of this object?

Topology is concerned less with the measurement of things, and more concerned with the structure of objects when you remove the notion of measurement entirely. In other words, what does it mean to be "near" something when you can't measure distance? What does it mean to be connected? Or having "essentially the same structure" when you disregard the shape?

 

[mathwonk]

I agree, the difference is measurement.

 

[Studiot]

I really appreciate the pictural help

Try these.

Draw a pentagon and inscribe a star,

Geometry tells you you have a pentagon and a star and all that stuff, that others have mentioned, about angles and sides and how they change if for instance you tilt the paper to view it at an angle.

Topology tells you that every vertex is connected to every other vertex.

Take a tennis ball.

Geometry tells you that you have a certain surface area, radius that every point on the surface is equidistant from the centre and so on.

Topology tells you that if you brush the hairs on the surface all one way there is at least one point where there is a complete reversal.

 

[Vargo]

The difference is that topology is a very well-defined subject. It is the study of continuity. In particular, you need a topological space and topological properties are the ones that are preserved under homeomorphism.

According to Wikepedia, Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. In other words it is vaguely defined and could be construed to include almost all of mathematics.

However, like others said, it is often understood that geometry involves some kind of measurement. Unfortunately, this ignores fields such as axiomatic projective geometry and ordered geometry which involve no measurement at all. Nowadays, relatively few people work in those areas though, so no one is likely to object if you exclude them. Really though, check out Desargues' theorem and Pappus' theorem on wikipedia and tell me those are not fundamentally "geometric".

 

[Vargo]

Let me just add that algebraic geometry is a huge field with a lot of practitioners. While I am not too familiar with the subject, I don't think measurement has a lot to do with it. What sort of measurement would be preserved by a birational equivalence?

 

[mathwonk]

of course overly simple answers to vague questions are always subject to modification. Algebraic geometry is in between geometry and topology. E.g. if we consider compact orientable surfaces of genus one, all of them are equivalent topologically since all are homeomorphic to a torus.

If we put locally Eucldean metrics on them, there are many different such "flat" metrics on them. Thus if we define sameness as "isometry", or having the same measure, there are a lot of different ones.

On the other hand, in complex algebraic geometry, we put an algebraic structure, which turns out to be equivalent to a holomorphic structure on them and then two are equivalent iff there is a holomorphic homeomorphism between them.

There are still a lot of them but not as many as there are isometry classes. I.e. this is not the same as there being an isometry, but it is close. Namely it is equivalent to there being a similarity between them, i,e, a homeomorphism that is an isometry except for a scale factor.

the point is that holomorphic/algebraic geometry does measure angles, if not lengths.

 

[Vargo]

Good point about vague questions. Thanks for clarifying that about algebraic geometry.

 

[lavinia]

Intuitively, for me, geometry means something with a specific shape, something you could potentially visualize. Topology does not assign a specific shape but rather asks what is the same among specific shapes that are continuously deformable into each other.

I do not know how to interpret conformal structures in these terms. Mathwonk, how do you do it?

It does seem though that each conformal structure on a surface corresponds to a shape that is the shape the surface would have with a metric of constant Gauss curvature equal to +-1 or zero in the case of a torus.

 

[mathwonk]

i'l try to say something later on conformal structures. for now i just read an realized how pertinent is micromass' answer.

 

[member 11137]

i'l try to say something later on conformal structures. for now i just read an realized how pertinent is micromass' answer.

Indeed.

And thank you very much for the different opinions and for the recommanded reference.