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I'm reading the section on transversality in Lee's Introduction to Smooth Manifolds. There, he characterizes the property of being the graph of a smooth map in a global sense (Theorem 6.32) and gives a criterion for the local sense (Corollary 6.33).

Is being the graph of a smooth map (whether in a global or a local sense) a desirable property?

Have this property interesting consequences?

I acknowledge that the questions are slightly open and would appreciate any answer.

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    $\begingroup$ Going all the way down to single-variable calculus, it's much easier to find the slope of a line tangent to a curve when the curve is the graph of a function. $\endgroup$ Commented May 13 at 13:45

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Probably the reason that such a result should resonate as important is that, after all, the implicit function theorem is also a theorem in the same essence: It proves that some things (there, they are the level sets of a map) are the graph of some functions, and there are many applications to this, as you have probably seen in the various proofs that use the implicit function theorem (in proof of the regular level set theorem, you use the resulting function to give charts about points in the pre-image of the regular value. In Lie groups, you use it to prove the inverse operation is smooth from knowing that the multiplication is smooth; etc.). Here also for example, you would know (in Lee's notation) that $S$ is diffeomorphic to the domain of the function i.e. $M$ (similar to implicit function theorem), which is already a good information. In this theorem it is proved that the diffeomorphism is given globally by the projection map.

Also for example, $N$ can sometimes be a vector space (e.g. the tangent space, so sometimes $M\times N$ is the tangent bundle); and in those cases, the result tells us that $S$ is a section of your bundle $M\times N$ (e.g. sometimes $S$ is equivalent to $f:M\to N$ which is a vector field on $M$). Now sometimes your vector bundles are not as simple as $M \times N$ for a vector space $N$, but using the local version of the theorem and the trivializations we have on any bundle, we can extend this to arbitrary bundles. In symplectic topology people (initially motivated by physics) are interested in studying lagrangian submanifolds $S$ of a symplectic manifold; these submanifolds are some of the "physically" well-behaved ones. And the base symplectic manifold sometimes just happens to be a cotangent bundle $T^*M$. So in this story for example, the theorem characterizes when we can think of a lagrangian submanifold as a differential 1-form over $M$ (which is a handy thing).

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