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When I started to learn Differential geometry, I wondered why a smooth manifold is defined to have an atlas that is not only smooth, but also maximal/complete.

Do you know of any theorems that rely on the fact that the atlas of a smooth manifold is maximal?

Edit: I am aware of the fact that a smooth atlas is contained in exactly one maximal smooth atlas, but I would be interested in an answer nevertheless.

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  • $\begingroup$ Every atlas is contained in some maximal atlas, so there is no loss of generality. $\endgroup$ – Zhen Lin Jan 5 at 9:35
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The maximality condition is included so that, in effect, anything that should be a coordinate chart is actually a coordinate chart. Here is an excerpt from Loring Tu's An Introduction to Manifolds which formalizes this.

Proposition 6.11. Let $U$ be an open subset of a manifold $M$ of dimension $n$. If $F \colon U \to F(U) \subset \mathbb{R}^n$ is a diffeomorphism onto an open subset of $\mathbb{R}^n$, then $(U, F)$ is a chart in the differentiable structure of $M$.

The proof is a straightforward application of the maximality assumption on the differentiable structure of $F$; for completeness, I have included it. (Proposition 6.10, referenced therein, just says that the coordinate charts are diffeomorphisms.)

Proof. For any chart $(U_\alpha, \phi_\alpha)$ in the maximal atlas of $M$, both $\phi_\alpha$ and $\phi_\alpha^{-1}$ are $C^\infty$ by Proposition 6.10. As composites of $C^\infty$ maps, both $F \circ \phi_\alpha^{-1}$ and $\phi_\alpha \circ F^{-1}$ are $C^\infty$. Hence, $(U, F)$ is compatible with the maximal atlas. By the maximality of the atlas, the chart $(U, F)$ is in the atlas.

Here is an example of Proposition 6.11 at work, again from Tu. You can think of it as a version of the inverse function theorem, if you'd like.

Corollary 6.27. Let $N$ be a manifold of dimension $n$. A set of $n$ smooth functions $F^1, \dots, F^n$ defined on a coordinate neighborhood $(U, x^1, \dots, x^n)$ of a point $p \in N$ forms a coordinate system about $p$ if and only if the Jacobian determinant $\det [\partial F^i / \partial x^j(p)]$ is nonzero.

The proof requires (a version of) the inverse function theorem on manifolds, so in order not to stray too far off topic, I have omitted it.

Another example of this in action is in nearly any theorem that has to do with the existence of local coordinates near a point. For example, the constant rank theorem, which states that under certain conditions on a function at a point, you can find coordinate charts near that point and its image in which the function has a particularly nice "normal form." (See Tu, Theorem 11.1, or John M. Lee's Introduction to Smooth Manifolds, Theorem 4.12.) Or, perhaps, in the theorem that any set of non-zero commuting coordinate vector fields are actually the coordinate vector fields for some chart. (See Lee, Theorem 9.46.)

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  • $\begingroup$ Great answer :) Thank you very much for including the references. I had the suspiscion that the "local slice criterion for embedded submanifolds" (Theorem 5.8 in John Lee's book) is only valid for a maximal atlas and ideed, the proof is based on the rank theorem. $\endgroup$ – Filippo Jan 7 at 12:33

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