What is geometry?

[ajb]

How would you define geometry as to encompass all it has become?

 

For me, something like the study of locally ringed spaces would be good. It includes manifolds, supermanifolds, schemes and NCG's.

 

To quote Manin,

 

that to do a geometry you do not need a space you only need an algebra of functions on this would-be space

 

 

What are you thoughts?

Edited January 3, 2010 by ajb

[michel123456]

For me geometry means proportion without the need of any absolute. Pure relativity. (not Relativity)

[ajb]

For me geometry means proportion without the need of any absolute. Pure relativity. (not Relativity)

 

 

No distinguished coordinates?

 

(They maybe distinguished classes of coordinates of course)

[Abstract_Logic]

I would define geometry as the study of the generalization of spatial structures and of operations on these spatial structures. This definition can include Euclidean spaces, non-euclidean, riemannian spaces, etc, as well as manifolds, supermanifolds, and other spatial structures.

[michel123456]

No distinguished coordinates?

 

(They maybe distinguished classes of coordinates of course)

 

Not sure what you mean by distinguished.

I may look old fashioned, but as an architect working with euclidian geometry all the time, I consider geometry as a relationship of shapes independently of any measurement. Always, in order to transform a geometric shape into an object (a building in my case), you have to input dimensionning artificially (the human scale as we say in my job). Pure geometry has no scale, no dimensionning, no absolute, no big, no small. Only relations. Pure relativity (not Relativity).

That's the reason why when I see theories or manifolds that take a specific value as a basic input I am highly suspicious.

[ajb]

I would define geometry as the study of the generalization of spatial structures and of operations on these spatial structures.

 

Not a bad way to view it, but not clear on what you mean by generalised spacial structures?

 

... Euclidean spaces, non-euclidean, riemannian spaces, etc, as well as manifolds, supermanifolds, ...

 

These examples can all be view as locally ringed spaces (with extra structures of metrics in the case of Riemannian spaces).

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Not sure what you mean by distinguished.

 

Thinking of manifolds in particular, there is in general no special coordinates. However, if the manifold comes with an extra structure then their maybe coordinates adapted to this structure.

 

For example, on Minkowski space-time one can use any coordinates you please. However, the preferred ones are the inertial observers. (There will in general also only be restrictions on the morphisms.)

 

 

I may look old fashioned, but as an architect working with euclidian geometry all the time, I consider geometry as a relationship of shapes independently of any measurement. Always, in order to transform a geometric shape into an object (a building in my case), you have to input dimensionning artificially (the human scale as we say in my job).

 

Shapes and geometric figures are very classical things. You can think of them as subsets of the set of points of your space. Geometry has come a long way past these things, but such things are what most people think of as "geometry".

 

 

 

Pure geometry has no scale, no dimensionning, no absolute, no big, no small. Only relations. Pure relativity (not Relativity).

 

Makes sense, but you can have a notion of distance on your space. Physical theories often rely on a scale, theories that don't care about scale are known as conformal. You do not always have a notion of distance/scale, nor in more generality do we have a notion of "closeness". That is the modern notion of a geometry is wider than just a topological space. Particularly true of noncommutative geometries, but is also true of classical sheaves and schemes.

 

 

That's the reason why when I see theories or manifolds that take a specific value as a basic input I am highly suspicious.

 

You should view things that rely on very specific coordinates as suspicious.

 

 

I am not sure how I would exactly describe geometry (locally ringed spaces seems ok to me), but generally we have an algebra which we treat as the algebra of functions on a "space". You then use geometric language and intuition to work on this algebra. Doing so can mean that points are an emergent idea and not the starting place. In fact, you can lose the notion of a point completely or realise that the geometry is not completely specified by the points.

Edited January 9, 2010 by ajb
Consecutive posts merged.

[michel123456]

For me the important thing in geometry is the absence of scale factor.

In think a thread about scale factor would be interesting. When Galileo discovered the importance of the scale factor in applicated euclidian geometry, he tooched a bit of devil's tail (or God's, if God has a tail).

But this is not mathematics. Out of the subject, sorry.

[ajb]

For me the important thing in geometry is the absence of scale factor.

 

In the context of Euclidean geometry a scale factor is a type is an affine map [math]\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}[/math].

 

I am not sure how you would propose to study spaces without their morphisms. (Of course a scaling is not a Euclidean morphism, i.e. it does not preserve the notion of distance .)

 

 

In a wider context of Riemannian geometry you have isometries and conformal transformations. The first preserves the metric (notion of distance) as where the second preserves angles but not lengths.

 

Without a metric you do not have the notion of scaling.

Edited January 9, 2010 by ajb

[michel123456]

In the context of Euclidean geometry a scale factor is a type is an affine map [math]\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}[/math].

 

I am not sure how you would propose to study spaces without their morphisms. (Of course a scaling is not a Euclidean morphism, i.e. it does not preserve the notion of distance .)

 

 

In a wider context of Riemannian geometry you have isometries and conformal transformations. The first preserves the metric (notion of distance) as where the second preserves angles but not lengths.

 

Without a metric you do not have the notion of scaling.

 

Exactly. If I recall well that was Ernst Mach who's quest was a universe without a metric. There are a few marginal followers, like J.Barbour but they encounter very serious problems with their unconventional search. see for info http://fqxi.org/community/articles/display/117

[ajb]

I don't know, but for sure Mach's ideas of inertial and matter as in Mach's principle are not realised in general relativity.

 

Gravity without a metric seems very odd to me. Well, even physical theories seem difficult to think about without a metric of some sort. You often require a method of creating scalers from various tensors. The most obvious way is contraction via a metric tensor. Also densities are required, they come naturally from a metric, though there is no reason not top pick an independent density.

 

What could be ok is to view the metric as an emergent thing. That is probably not a bad idea to pursue. I think in some approaches to quantum gravity this is the approach.

 

But anyway, a metric is an additional structure in the theory of (super)manifolds. You do not need such a thing. Indeed, I have only touched upon the theory of metrics on manifolds in my studies. It is very interesting.

 

What is true is that every manifold can be equipped with a Riemannian metric (for a Lorentzian one there is a topological obstruction). You can do this by picking a partition of unity and then gluing the Euclidean metrics on each open set to crate a global metric. But this is not canonical.

Edited January 9, 2010 by ajb

[michel123456]

I don't know, but for sure Mach's ideas of inertial and matter as in Mach's principle are not realised in general relativity.

 

Gravity without a metric seems very odd to me.(...)

What could be ok is to view the metric as an emergent thing. (...)

What is true is that every manifold can be equipped with a Riemannian metric (for a Lorentzian one there is a topological obstruction). You can do this by picking a partition of unity and then gluing the Euclidean metrics on each open set to crate a global metric. But this is not canonical.

 

Could you please explain your last paragraph? Sounds terribly interesting.

[ajb]

Could you please explain your last paragraph? Sounds terribly interesting.

 

I may be a little sketchy here.

 

Let us work in the smooth category. Let [math]M[/math] be a smooth manifold. (I assume to be paracompact by definition). Consider an atlas

 

[math]\{ U_{\alpha}, \phi(U_{\alpha}) \}[/math]

 

[math]\phi: U_{\alpha} \rightarrow \phi(U_{\alpha}) \subseteq\mathbb{R}^{n}[/math]

 

which is locally finite. That is each point has a neighbourhood that intersects only a finite number of the open sets in the atlas.

 

Subordinate to this we have a differentiable partition of unity [math]\tau_{\alpha}[/math]. Then define

 

[math]g = \sum_{\beta} \tau_{\beta}\phi^{*}_{\beta}g_{Euclidean}[/math]

 

This defines a metric on the manifold.

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Another influential idea in geometry was Klein's Erlanger Program, where geometry is "the study of invariants of group actions on sets".

 

In the widest context I guess this could mean looking at (local?) automorphism of a ring/algebra and defining geometry in that way. I'd need to read up on this before passing comment.

[the tree]

I've been thinking about the original question, there was a lot of technical language in your proposed answer and TBH I don't even understand what a sheaf is.

 

To me, you start doing geometry once you involve a meaningful notion of distance: for it to be meaningful there should be a distance function where distances between distinct pairs can be equal but aren't always - that way you can have shapes and congruences that you'd expect from something called geometry. Of course that may be a really simplistic definition but it covers all the things I've known to be called geometry.

[ajb]

To me, you start doing geometry once you involve a meaningful notion of distance: for it to be meaningful there should be a distance function where distances between distinct pairs can be equal but aren't always - that way you can have shapes and congruences that you'd expect from something called geometry.

 

Geometry comes from the Greek geo- "earth", -metria "measurement" so geometry = " earth measurement".

 

This is in essence what you describe. A notion of a "space" and some "measurement".

 

Of course that may be a really simplistic definition but it covers all the things I've known to be called geometry.

 

It does indeed cover what one usually thinks of as geometry.

 

Though in modern geometry, for example studying topological manifolds we only have the property of locally metrization. This is because topological manifolds locally look like Euclidean space, including the metric. Generically we do not need the notion of a distance to discuss other geometric objects on manifolds (thinking of smooth manifolds in particular).

 

Anyway, I use geometry all the time, I may even call myself a geometer. However, I have not found a completely satisfactory definition of what geometry is.

Edited January 22, 2010 by ajb

[Royston]

It does indeed cover what one usually thinks of as geometry.

 

Though in modern geometry...

 

I think that's the problem with this question, modern geometry is a rather specialized field, and you're probably the best equipped to answer that question ajb. However, I've just started studying GR, so I'll give my opinion based on a basic understanding.

 

AFAIK, the insight that has lead to modern geometry, would be Gauss' 'remarkable theorem', which did away with a coordinate system. Following that would be Riemannian geometry, where I guess a metric went beyond unit intervals (i.e the distance between each unit changes), so there's no such notion of distance...I know there's a lot more to it than that.

 

My point being, geometry was a completely different thing up until the 19th century, and I'm sure, hitherto at your level, far more abstract now. I'm sure it would be a hard thing to define, unless you really knew the subject.

[Tau Meson]

Anyway, I use geometry all the time, I may even call myself a geometer. However, I have not found a completely satisfactory definition of what geometry is.

 

 

Well, I guess I would have to ask is what is wrong with having different definitions of geometry? For the average person, the only thing that they are vaguely familiar with is Euclidean space, and that is the definition that everyone ends up using.

 

I'm just beginning to learn what Riemann geometry and Topology is, but I think the normal definition of geometry for these spaces don't quite cut it. I am more inclined to describe them using graph theory or mappings.

[ajb]

Well, I guess I would have to ask is what is wrong with having different definitions of geometry?

 

Nothing. As subjects evolve they will almost inevitably out grow their "original definitions".

 

To a large extent the question of philosophical. Geometry is what geometers do!

 

I'm just beginning to learn what Riemann geometry and Topology is, but I think the normal definition of geometry for these spaces don't quite cut it. I am more inclined to describe them using graph theory or mappings.

 

Mappings are fundamental in modern geometry. Roughly, I think of geometry as "algebra" in which we need to think about "coordinate changes".

 

Category theory is an important part of modern mathematics and geometry. Here we think of the maps between objects as being the fundamental things as opposed to the objects themselves. You inclination to use graphs and mappings sounds like category theory.

[khaled]

my answer is simple,

 

Geometry is the Visual version of Algebra