I read somewhere that radial co-ordinates are not considered charts for n-spheres and that a minimum of n charts are required to map it properly.
With regards to the suitability of radial co-ordinates as charts I have noted the following points:
-As the n-sphere has a pre-defined metric, there is already a definition of distance and 'closeness'.
-According to the book, due to discontinuities at some places (eg: the international date line in earth's case) and singularities at some points, radial co-ordinates are unsuitable.
-Perhaps I have not clearly understood, but don't the co-ordinate discontinuities and singularities arise due to the pre-defined metric on the n-sphere?
So from this can I conclude the following?
-To decide the minimum number of charts required to completely cover the manifold, we need a definition (like for the n-sphere).
-In this definition (for example x^2 + y^2 = a, in the case of a 1-sphere), we need to use an arbitrary co-ordinate system as a 'crutch'.
-Thus, to figure out the minimum required charts to cover the manifold (in manifolds like the n-sphere) we need to use an arbitrary co-ordinate system to first 'introduce' us to the manifold.
Best Answer
This is false, unless there's a hidden requirement in "properly". Two charts suffice: stereographic projection from the North pole, and similar projection from the South pole.
Yes, before we say anything about some object, we need to define it. For example: unit $n$-sphere is the set of unit vectors in $\mathbb R^{n+1}$ with the induced topology.
If we are defining a manifold in terms of a larger manifold containing it, having a coordinate system on that larger manifold certainly helps. We have to describe things in terms of other things we already know, and we all know $\mathbb R^{n+1}$ with its standard coordinates.
That said, manifolds are not always defined by giving some equation, or set of equations in $\mathbb R^{n+1}$. For example, the sentence let $N$ be the tangent bundle of $S^2$ defines a four-dimensional manifold without referring to any Euclidean space containing it.
Well, if you put any other metric on the surface of Earth, we would still have the dateline, would not we? A sphere cannot be covered by a single chart because it's not diffeomorphic to any Euclidean space. In other words, it's because the sphere is not a boring flat manifold like a piece of a plane. This is not some deficiency of the spherical metric.