I am learning in differential geometry that in order to implement differential caluclus into a manifold $M$ you need to define coordinate charts between the manifold $M$ and $R^n$ satisfying certain properties and such that the transition between two such charts is smooth (understood as a differentiable function in $R^n$.
I can't find anywhere in a textbook a reason as to understand, why one needs to transfer notions of differential calculus on $M$ into notions of differential calculus in $R^n$. Is this a necessity or a (comfortable ) choice ? Is there a differentiability concept on $M$ possible without any reference to $R^n$?
In general relativity books one reads sometimes that it is possible to treat the corresponding manifolds (pseudo-rimannian geometry) so as to avoid coordinates. I am wondering how in this case one would apply the differentiable calculus following the scheme described above ?
In the book ''Differential Geometry'' by S.S. Chern, at the end of the book there is a historical excursion out of which the conclusion following a citation is: ''It is not easy to free oneself from the idea that coordinates have immediate metrical meaning'' and that ''coordinates are meaningless....''. At the other hand, to define a metric you necessarily need coordinates. So, i am totally confused! Can somebody say a word on this ? Can one give a comment on the two citations provided and explain what is exactly meant by them?
I will appreciate any answer concerning the questions on the 3 points above. Thanks.
The coordinate charts are needed not only to define a calculus on manifolds, but to define the manifolds themselves. If you have a topological space and you want to show that it is a smooth $n$-dimensional manifold, you have to show that for every point in the space there is an open neighborhood that is diffeomorphic to an open set in $\mathbb{R^n}$. The diffeomorphic maps from the manifold to $\mathbb{R^n}$ are called charts.
In ordinary (multivariable) calculus one needs to define things like $f(x+h)-f(x)$, which is only possible if the domain of the function is a vector space. The charts allow us to transfer the notion of differentiability by pulling back the function along the chart to a function $\mathbb{R^n}\rightarrow\mathbb{R}$, for which differentiability is defined. If we have two charts, the fact that going from one chart to the other is diffeomorphic means that the differentiability on the manifold is well-definied (does not depend on the chart).
When one says that one can 'avoid coordinates', it usually (see the caveat mentioned in the comment below) means the properties of objects that live on the manifold do not depend on specific coordinate charts. Thus any meaningful results should be expressed in a formalism that does not make reference to specific coordinate charts. Needless to say, concrete examples or calculations often require the choice of a specific coordinate chart. The important thing is that in principle any coordinate chart works. For example, if we define the metric using one set of coordinates, we can then express it in any other set of coordinates without changing the metric itself. Thus, geometrical statements about this metric, such as the arc lengths of curves, do not depend on the choice of coordinates.