What is the concept of a natural coordinate in manifold geometry?

[ShayanJ]

I'm just learning manifold geometry and tensor analysis.From the things I've understood till now,an idea came into my mind but I can find it or its negation no where.So I came to ask it here.
I can't explain how I deduced this but I think there should be sth like a natural coordinate for a particular manifold.I mean sth that is the most suitable or maybe the only possible map for it to make an atlas.E.g. Cartesian coordinates for euclidean manifold and spherical polar coordinates for a spherical manifold.
And I should tell I've not even finished one book on manifold geometry so if I'm telling sth crazy here,I apologize.
Thanks in advance

 

[Fredrik]

For some manifolds, there are coordinate systems that are "special" in some sense. But I don't think that anything like that holds in general.

 

[quasar987]

There is no such thing as a canonically given atlas for any manifold Shyan, but there are some manifolds that come up often, like R^n, the circle, tori, CP^n on which there is an obvious choice of atlas. So much so that people won't bother specifying what atlas they are working with in these case, or they might refer to it as "the natural atlas". But it is only natural in the sense that it is convenient and widely used.

 

[ShayanJ]

thanks for the answers.
So could you take a look at https://www.physicsforums.com/showthread.php?t=585238 .
thanks again

 

[blue_raver22]

I am happy to see that you are exploring and trying to understand manifold geometry and tensor analysis. It is a complex and fascinating field that has many applications in physics, engineering, and mathematics.

Firstly, I want to clarify that there is no such thing as a "natural coordinate" for a manifold. A manifold is a mathematical concept that represents a space that is locally similar to Euclidean space. It can have different shapes and structures, and therefore, there is no one specific coordinate system that is suitable for all manifolds.

In fact, one of the key ideas in manifold geometry is the concept of coordinate independence. This means that the laws and equations that describe a manifold should not depend on the choice of coordinates used to describe it. Therefore, any coordinate system can be used to map out a manifold, as long as it is consistent and covers the entire space.

That being said, certain coordinate systems may be more convenient or intuitive for certain manifolds. For example, Cartesian coordinates are often used for Euclidean manifolds because they are simple and familiar, while spherical polar coordinates are more suitable for spherical manifolds because they reflect the spherical symmetry of the space.

I applaud your curiosity and encourage you to continue learning about manifold geometry. It is a vast and complex subject, and it takes time and effort to fully understand it. Don't worry if you feel like you are not grasping everything right away – it takes time and practice to develop a deep understanding of any scientific concept. Keep asking questions and seeking out resources, and you will eventually master this fascinating field.