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These definition are as given in a book. I need some clarification of doubts on this topic.

Definition : Let $ \mathcal{A} = {(x_{\alpha},U_{\alpha})}_{\alpha \in A}$ be an atlas on a topological manifold M. Whenever the overlap $U_\alpha \cap U_\beta$ between two chart domains is nonempty we have the change of coordinates map $ x_\beta \circ x_\alpha^{-1} : x_\alpha(U_\alpha \cap U_\beta) \to x_\beta(U_\alpha \cap U_\beta)$. If all such change of coordinates maps are $C^r$-diffeomorphisms then we call the atlas a $C^r$-atlas.

Definition: A maximal $C^r$-atlas for a manifold $M$ is called a $C^r$-differentiable structure. The manifold $M$ together with this structure is called a $C^r$-differentiable manifold.

In the first definition what is meant by change of co-ordinates map ? What is the intuition behind this ? How is it exactly done. Why is the change of co-ordinate map needs to be a $C^r$ diffeomorphism ? Diffeomorphism is a mapping from one differentiable manifold to another with some properties. How can one use the term 'diffeomorphism' while defining a differentiable manifold ?

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You first define differentiability for maps between open subsets of $\mathbb R^n$. Then you use this to define differentiable manifolds and differentiable maps between them. –  Rasmus Apr 10 '11 at 9:16

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In the first definition what is meant by change of co-ordinates map ? What is the intuition behind this ?

A simple example is the change from cartesian coordinates in $\mathbb{R}^2$ to polar coordinates. You start with one specific coordinate system = map, cartesian (x, y)-coordinates with respect to a fixed point of reference that is described by (0, 0), and you change coordinates to polar coordinates $(r, \phi)$ acoording to

$x = r \; cos(\phi)$

$y = r \; sin(\phi)$

Why is the change of co-ordinate map needs to be a $C^2r$ diffeomorphism ?

That's just a part of the definition, if all coordinate changes are r-times differentiable, we have a $C^r$ - manifold.

Diffeomorphism is a mapping from one differentiable manifold to another with some properties. How can one use the term 'diffeomorphism' while defining a differentiable manifold ?

A diffeomorphism is a bijective map from $\mathbb{R}^n$ to $\mathbb{R}^n$ whose inverse function is also differentiable. This definition of a "diffeomorphism" does not use the concept of a manifold. The basic strategy when you handle manifolds, is to reduce the question/definition/whatever at hand to a question involving $\mathbb{R}^n$ only, by applying charts. So, a diffeomorphism $\psi$ of differential manifolds $M \to N$ is defined to be a map such that the combination with charts $\phi_n \psi \phi_m^{-1}$ is a diffeomorphism for all charts $\phi_n$ of N, $ \phi_m$ of M.

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A change-of-coordinates map takes a set of coordinates in the $\alpha$ system and outputs the $\beta$ coordinates by using the inverse of the $\alpha$ chart function $x_\alpha$ to take the coordinates in $x_\alpha(U_\alpha)$ and get the corresponding point on $U_\alpha$, which if also a point on $U_\beta$ can therefore be represented by coordinates in the $\beta$ system using the $\beta$ chart function $x_\beta$; the single function representing this change of coordinates is therefore $x_\beta\circ x_\alpha^{-1}$

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