Deciding whether an atlas is smooth

differential-geometrysmooth-manifolds

Let us begin with some definitions regarding smooth atlases. They are quoted from the book by John M. Lee.

Let $M$ be a topological $n$-manifold. If $(U,\varphi)$, $(V,\psi)$ are two charts such that $U\cap V\neq\emptyset$, the composite map $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$ is called the transition map from $\varphi$ to $\psi$. It is a composition of homeomorphisms, and is therefore itself a homeomorphism. Two charts $(U,\varphi)$ and $(V,\psi)$ are said to be smoothly compatible if either $U\cap V=\emptyset$ or the transition map $\psi\circ\varphi^{-1}$ is a diffeomorphism. Since $\varphi(U\cap V)$ and $\psi(U\cap V)$ are open subsets of $\mathbb{R}^n$, smoothness of this map is to be interpreted in the ordinary sense of having continuous partial derivatives of all orders.

We define an atlas for $M$ to be a collection of charts whose domains cover $M$. An atlas $\mathcal A$ is called a smooth atlas if any two charts in $\mathcal A$ are smoothly compatible with each other.

To show that an atlas is smooth, we need only verify that each transition map $\psi\circ\varphi^{-1}$ is smooth whenever $(U,\varphi)$ and $(V,\psi)$ are charts in $\mathcal A$; once we have proved this, it follows that $\psi\circ\varphi^{-1}$ is a diffeomorphism because its inverse $(\psi\circ\varphi^{-1})^{-1}=\varphi\circ\psi^{-1}$ is one of the transition maps we have already shown to be smooth. $\color{red}{Alternatively}$, given two particular charts $(U,\varphi)$ and $(V,\psi)$, it is often easiest to show that they are smoothly compatible by verifying that $\psi\circ\varphi^{-1}$ is smooth and injective with nonsingular Jacobian at each point, and appealing to Corollary C.36.

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I wonder why the alternative way is considered to be often easiest by the author. It seems like no work is saved. Worse, we take more steps by looking at injectivity, Jacobian, etc. Based on the discussion prior to the alternative method, if $\psi\circ\varphi^{-1}$ is shown to be smooth, we are done with checking that the given atlas is smooth. Did I miss anything?

Best Answer

The paragraph is referring to two different (albeit closely related) procedures:

  • To show an atlas is smooth, it suffices to show that every transition map $\psi \circ \phi^{-1}$ is smooth.
  • To show $(U, \phi)$ and $(V, \psi)$ are smoothly compatible, i.e. that $\psi \circ \phi^{-1}$ is a diffeomorphism, it suffices to show that $\psi \circ \phi^{-1}$ is smooth + injective + nonsingular Jacobian. This saves us the trouble of verifying that the inverse map $\phi \circ \psi^{-1}$ is smooth.
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