Do we loose information when we project a manifold.
For example, do we loose information about the manifold i.e. Earth (Globe) when we project it to a chart in the book (using maybe stereographic, mercator or any other method)
Similarly, we should be loosing information while we create a Bloch sphere for a 2 state system in Quantum Mechanics which is also a Projected space from a higher dimension i.e. 4 dim.
Also, is there a way to quantify this information loss, if there is any?
Dear Chetan Jesing Waghela ,
In general, the answer is yes, we will lose information, because projections are not necessarily homeomorphisms.
And in this case, if P is a projection from M to N, then
M ⊑ P-1 (P(M)) and P( P-1 (N)) ⊑ N.
To prevent such case we use atlases for the manifold M which includes
charts ( local representation of M ).
So, we don't project the whole sphere ( Globe) at once this will cause the loss
of information ( in your language), simply because there is no single homeomorphism that projects the whole sphere at once into the plane.
We overcome this situation by dividing the sphere into parts ( called charts)
and then determine a family of ( homeomorphisms ) act as projections at the same time. This will help to preserve the information.
Best regards
Issam Kaddoura Is this the reason the field of Differential Geometry strives to have an intrinsic definition and tries to avoid embeddings?
Dear Chetan Jesing Waghela ,
Embeddings or imbeddings are continuous injective mappings.
They are very useful to study manifolds. we embed manifolds of dimension m in manifolds or spaces of higher dimension say n , where m ≤ n.
So, you can study the ( surface of) the sphere S 2 that has dimension 2 and embed it in R3 ( The three - dimensional Euclidean space ).
i.e, there is an embedding from S2 into R3.
We need embeddings to study manifolds and don't avoid them.
Best wishes
Issam Kaddoura Maybe you didn't understand my comment. I am trying to say that usually textbooks in Physics mention that the whole exercise to create the language of differential geometry and flows is to have an intrinsic definition of manifolds and vectors.
I am asking the reason to have an intrinsic definition is that we loose the information when we create charts (which is an extrinsic viewpoint)?
Thanks for the patience.
Dear Chetan Jesing Waghela ,
Yes you are right, in mathematical models that describe the real phenomenon in our surrounding concerning the continuum movements of particles , the using mappings should diffeomorphisms or at least homeomorphisms to preserve
the model very close to the real situation.
Best regards
@Issam Kaddoura Thanks ! I understood what you are saying. If the transformation used to charting is not a diffeomorphism (I.e. invertible) then we loose information.
Thanks.
when we project (Earth) to a chart in the book, if the transformation is diffeomorphism we will have a minor copy of the earth, some details may not be clear to be seen in the graph but this does't mean they are lost.
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