Theorems that rely on the fact that the atlas of a smooth manifold is maximal/complete

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.

• Every atlas is contained in some maximal atlas, so there is no loss of generality. – Zhen Lin Jan 5 at 9:35

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.)

• 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. – Filippo Jan 7 at 12:33