A What Is the Gauss Embedding Theorem Mentioned in the Video Talk?

The Tortoise-Man
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Does anybody know to which "Gauss embedding theorem" the speaker in this video talk at minute 14 (point 5. in the displayed notes) is refering too? Sounds to be a standard result in differential geometry, but after detailed googling I found nothing to which the speaker may refering too in the linked video. Any ideas?
 
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The Tortoise-Man said:
Any ideas?
Nash embedding theorems?
 
I'm a bit confused by the conditions on the existence of coordinate basis given by Frobenius's theorem. Namely, let's take a n-dimensional smooth manifold and a set of n smooth vector fields defined on it. Suppose they are pointwise linearly independent and do commute each other (i.e. zero commutator/Lie bracket). That means they span the entire tangent space at any point and since commute, they define a local coordinate basis. What does this mean? Well, starting from any point on the...

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