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Let $M_c=\{(x_1, x_2, x_3, x_4)\in \mathbb{R}^4 \ |\ x_1x_2+x_2x_3+x_3x_4=c\}.$

Prove that $M_c$ is diffeomorphic to $\mathbb{R}^2\times S^1$ if $c\neq 0$ .

I tried to get a diffeomorphic mapping, but I couldn't think of anything.

Please tell me how to get the map.

Rosalina
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  • Have you tried diagonalizing the quadratic form? Find the eigendecomposition of its matrix. – coiso Aug 07 '23 at 15:36
  • No. I don't know how that is related to the question I asked. Please tell me more details. – Rosalina Aug 07 '23 at 15:46
  • By diagonalizing, you can write $x_1x_2+x_2x_3+x_3x_4=\lambda_1u_1^2+\lambda_2u_2^2+\lambda_3u_3^2+\lambda_4u_4^2$ in new coordinates, where $\lambda$s are the eigenvalues. You ought to end up with two positive and two negative eigenvalues, so you can rescale the coordinates to get $v_1^2+v_2^2-v_3^2-v_4^2$. Do you follow the reasoning up to that point? Do you know how to show the level sets of that are diffeomorphic to $S^2\times\Bbb R^2$? – coiso Aug 07 '23 at 15:54
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    Your question is a bare problem statement with no context. Let me suggest that you look at our guidelines for how to ask a good question, with emphasis on avoiding "I couldn't think of anything" questions, and on providing context. – Lee Mosher Aug 07 '23 at 16:05
  • Why are eigenvalues two positive and two negative? I thought the matrix of the quadratic form is \begin{pmatrix} 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ \end{pmatrix} $$ – Rosalina Aug 07 '23 at 16:19
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    @Rosalina That is one such matrix, but you ought to use a symmetric matrix because the form is symmetric and spectral theory applies to symmetric matrices. Average that matrix with its transpose. – coiso Aug 07 '23 at 16:30
  • @Lee Mosher I am very sorry. This question is the entrance exam to Tokyo Institute of Technology (graduate school) in 2020. I take the exam this year. Any friends couldn't solve this problem, so I want you to help me. – Rosalina Aug 07 '23 at 16:31
  • The duplicate question I have found has an interesting connection with this question – Jean Marie Aug 07 '23 at 23:10

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