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I'm studying for a differential geometry midterm and am stuck on what should be a pretty simple question. It asks for the solution to a system of equations as a one dimensional submanifold of $\mathbb{R} ^3. $ They are $$ x + y + z = 1 $$ and $$ x^4 +y ^4 + z ^4 = 3 $$

I get that there is 1 degree of freedom, so it makes sense that is is 1 dimensional, and I know the Hausdorff and countability condition for manifolds, but it seems like there's a theorem about sub-manifolds I am forgetting?

Would I prove it using some immersion or something from a compact set, like here , but the inclusion is not clear to me.

Andrew Hardy
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  • Do you know transverality? – Randall Feb 24 '19 at 03:53
  • I have a vague understanding, but it is not covered in the class notes. I am using the professors notes which they assigned instead of a textbook – Andrew Hardy Feb 24 '19 at 03:55
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    Have you folks learned to apply the implicit function theorem? This is the case where you have a mapping $F\colon\Bbb R^3\to\Bbb R^2$, and you're looking at $F^{-1}\big((1,3)\big)$. Does the derivative matrix have rank $2$ at every point of that set? – Ted Shifrin Feb 24 '19 at 05:13

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My attempt at a solution is to compute the Jacobian as \begin{bmatrix} 1 & 1 & 1 \\ 4x^3 & 4y^3 & 4z^3 \end{bmatrix} which is rank 2 because all the variables being zero is not a solution to the system. Also if x,y,z are the cube root are $ 1/4 $ they don't satisfy the system. This means I can use Regular Value Theorem ( which I think is what Ted Shifrin meant by applying Implicit Value Theorem ) to say that the dimension of the level set is 1 and that it is a submanifold.

I think I have covered everything now.

Andrew Hardy
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