Understand differentiable manifolds

[goksen]

I am trying to understand differentiable manifolds and have some questions about this topic:

We can think of a circle as a 1-dim manifold and make it into a differentiable manifold by defining a suitable atlas. For example two open sets and stereographic projection etc. would be the choice.But in anyway we have to begin with a coordinate system (and generally it is cartesian coordinate system ) Is there a way to assign charts to a circle without referring to cartesian coord.? And isn't that unusual to be able to define differentiablity without using metric, norm..

(I know my question looks silly , but I hope the replies will make some points more clear for me)

 

[goksen]

I want to add a familiar question of PF:

Why has'nt anybody answered my question?

A) the question is not clear
B)this is not the right forum
C)the title is not attractive

 

[mrandersdk]

Ok not sure why you talk about cartesian coordinates? A manifold is something that locally looks like eucidean space, is that wat you meen?

It is important to understandt that a manifold can be many crazy structures, such as fx. some collection of operators. So what would we meen by a operator being differential? We do not have some apparent notion of it, but we know what diff meens in euclidean space, so by requiring that or charts overlap smooth (you know what this meens?), we can thínk of every function from our manifold to fx R, as a map from R^n to R (assuming n dim manifold), where we know what diff meens.

So making something into a diff manifold is to give it a struture so that we locally can use what we know from euclidean space, which has a norm, and a nice topology and so on. But because it only locally looks like euclidean a lot of strange things can happen, which is what you study in manifold theory.

So to return to your first question, when thinking about the cirkel, you need to make charts that goes from S into R, by chart we meen that it need to be a homeomorphism from a open set of S to R. It is important to note that even though you can draw the circle in R^2, diff manifold does not meen that your chart

f: S subset of R^2-> R

is differential, but that all charts you construct must overlap smooth, which is very different (do you understand this differens?).

I understand why it seems like you already have coordinates because you construct the chart from thinking of the circle lying in R^2 and then making fx stereographic projection.

Maybe it who be a good exercise to try to think of the circle as the quotient space R/2pi*Z, and try to make charts to this. Or trying a more difficult manifold (it can help to take something more dificult because you don't place it R^n as easily), you could try to make the space of rotation in R^3 into a manifold.

that is you could represent a rotation by R(phi,n) where n is a unit vector (the rotation axis) and phi is the rotation angle.

http://en.wikipedia.org/wiki/Rotation_group

you need to make some clever maps to make them continous, if you give this space the induced topology from the operator norm

http://en.wikipedia.org/wiki/Operator_norm

on reason is that this space is isomorph to the closed unit ball in R^3, where you take oposite point on the boundary to be eqievalent, so you could try to start with making the space of closed unit ball where oposite points are equivalent to a manifold, then you will see that even though it is easy so picture it in R^3, the open sets are not like the open sets in the normal closed unit ball.

In fact you can't make the closed unit ball to a manifold (because it is closed), but this weirs costruction you can.

I stop now, it is a big subject so just keep reading and keep making exercises they will give you a feeling for it eventually

 

[wofsy]

A person on a manifold thinks he is in Euclidean space. Why? because he can construct an atlas which maps out his local area. There is nothing unique about his map though and he could just as well have mapped it in some other way. He is interested in how to compare coordinates on his two maps and so needs some rule which translates one map into the other. If this translation is differentiable then he can translate not only coordinates but vectors (velocities for instance) and tensors as well.

Now he gets ambitious and decides to cover his whole world with maps. Each area has a cooordinate map and for each overlapping pair he knows how to translate coordinates. He has described his world as a differentiable manifold. Notice that he has no idea what he world looks like globally. Everywhere he goes it just looks like another little Euclidean region to him.

If you think about it, you see that each if his maps is just a rule which associates a Euclidean coordinate to each location. This is what maps are. So he he has made a homeomorphism from this region into a region of Euclidean space. For instance steroegraphic projection is just the usual polar map that you can buy in a map store.

In order to do calculus on the manifold it is necessary to be able to compare derivatives on overlapping regions. This is why the overlaps need to be differentiable.