Size Manifold Globally: Euclidean 3D Space

[sqljunkey]

I was wondering if it was possible to determine the size of a manifold globally. Suppose I had a manifold that sits in 3 dimensions. I could construct a Euclidean space around in the same space and be able to say things of the dimensions right?

 

[fresh_42]

Your question doesn't make much sense. Manifolds are usually not embedded in some Euclidean surrounding space, which is why we consider them. Nevertheless there are some which can be seen as embedded in one. Then you have a geometrical object in a Euclidean space, which means that you can apply the language of geometry, and size and dimension whatever this means. A manifold is something which is locally Euclidean, like an atlas is flat whereas Earth is not. Its dimension is that of this Euclidean space: two dimensional in case of the Earth's surface, which is all we make an atlas of. Size almost necessarily requires a measurement from outside, i.e. an embedding in a Euclidean space. Then it's the size of e.g. a sphere or a torus. However, size is not a mathematical property of manifolds.

 

[sqljunkey]

Taking the example of the spherical Earth what if instead of constructing atlases and charts in 2d space we construct these in 3d space? So we would have charts of cities, rivers, oceans and the Earth's core etc all in 3d.

If i pieced this all together again would i get what i started with? So if i knew the dimensions of each 3d chart let's say wouldn't i be able to know the dimensions of the whole earth. ( if Earth can be considered a manifold)

 

[WWGD]

Taking the example of the spherical Earth what if instead of constructing atlases and charts in 2d space we construct these in 3d space? So we would have charts of cities, rivers, oceans and the Earth's core etc all in 3d.

If i pieced this all together again would i get what i started with? So if i knew the dimensions of each 3d chart let's say wouldn't i be able to know the dimensions of the whole earth. ( if Earth can be considered a manifold)

But since the Earth is locally 2D , topological properties will be distorted in 3d charts. Unless maybe you make a copy/replica of the Earth in 3-d. There is such a think as area forms and volume forms but these are more geometric than topological, i.e., they are not invariant under topological transformations.
Edit: and, yes, there is the information in the overlap maps that allows you to glue together the pieces into the original.

 

[WWGD]

I was wondering if it was possible to determine the size of a manifold globally. Suppose I had a manifold that sits in 3 dimensions. I could construct a Euclidean space around in the same space and be able to say things of the dimensions right?

What is the dimension of the manifold itself? This is an invariant. And what other dimensions are you referring to?As an example, A circle is 1-dimensional but it may sit in 2,3 or higher dimensions.

 

[sqljunkey]

What is the dimension of the manifold itself?

I assumed 3-D.

But since the Earth is locally 2D , topological properties will be distorted in 3d charts.

I didn't understand when you said locally flat.

If I brought a spoon today and started digging scoops of Earth one at a time, careful not to deform it's shape and I counted each scoop, one by one until I dug up the whole earth, wouldn't I be able to at least approximate the volume of the whole Earth by adding the number of scoops I dug?

You are somehow saying that even though I piece these 3-d scoops back together into the manifold I wouldn't have learned anything of the geometry because it could have had a higher dimension.

So, if I had a manifold with n dimension my charts should at least be n -1 ?

note my spoon measures volume in cubic cm.

 

[fresh_42]

... wouldn't I be able to at least approximate the volume of the whole Earth by adding the number of scoops I dug?

This doesn't work in the manifold picture. Locally flat means there is a certain kind of bijection, but it doesn't mean no deformation takes place. So in case of a three dimensional manifold when you cut out small pieces, transform them into a Euclidean space where size makes sense, you can add up those volumes, but it doesn't reflect the original size. And there is a technical difficulty: the local pieces of manifolds overlap. You cannot chose a closed part of them. Anyway, you can get an approximation, but of what? Volume isn't properly defined, it cannot be done. Except we have an embedding in a Euclidean space and we use its definition of volume. And this is geometry. The language of manifolds isn't necessary in this case.

In other words: In order to make sense of size or volume, we will have to add so many restrictions on the situation, that it is no longer a property of manifolds, but a property of geometric objects. You may still call those objects manifolds, but that would be like telling kids at school, that their fractions they calculate with aren't rational numbers but complex numbers instead. You can do this, but does it make sense?

 

[WWGD]

I assumed 3-D.
I didn't understand when you said locally flat.

If I brought a spoon today and started digging scoops of Earth one at a time, careful not to deform it's shape and I counted each scoop, one by one until I dug up the whole earth, wouldn't I be able to at least approximate the volume of the whole Earth by adding the number of scoops I dug?

You are somehow saying that even though I piece these 3-d scoops back together into the manifold I wouldn't have learned anything of the geometry because it could have had a higher dimension.

So, if I had a manifold with n dimension my charts should at least be n -1 ?

note my spoon measures volume in cubic cm.

I was thinking of the surface of the Earth which is locally 2D. The Earth with its interior is locally 3D.

 

[sqljunkey]

You may still call those objects manifolds, but that would be like telling kids at school, that their fractions they calculate with aren't rational numbers but complex numbers instead. You can do this, but does it make sense?

Well those kids might someday find this useful in their lives. :3

But I understand what you are saying.

But here is where it all gets bunched up in my head. Suppose I cut all these pieces, find out all these dimensions and then put them back together. I also now know how these pieces are interconnected. And I ask the question what is the shortest distance between the north pole and south pole.

By knowing how the pieces are interconnected and the size of each piece I can add all possible combinations and find the one that has the least "distance" from south pole and north pole. Wouldn't this be a global-like measurement of the manifold?

Meaning that I'm not embedding the manifold in a surrounding geometry, I'm using the information of the manifold itself, the way the charts are interconnected and the charts to find global measurements of the manifold.

(last reply :3)

 

[jbriggs444]

By knowing how the pieces are interconnected and the size of each piece I can add all possible combinations and find the one that has the least "distance" from south pole and north pole. Wouldn't this be a global-like measurement of the manifold?

That sounds eminently reasonable.

Perhaps, the least upper bound on the set of lengths of geodesics within the manifold. Which would be a bit greater than the pole to pole distance because a path following the equator halfway around is a geodesic and is a bit longer.

Actually, better tighten that up a bit -- the least upper bound on the set of lengths of the shortest geodesics between pairs of points within the manifold. With a requirement that there is at least one geodesic connecting each pair of points. [That eliminates the possibility of measuring the distance between Dallas and Forth Worth the wrong way around and getting a distance of 25,000 miles]

 

[WWGD]

Maybe you're also interested in Length Spaces?