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LetAt $x\in F^{-1}(-1)$,This differential is nonzero because $x\ne 0$, so $-1$ is a regular value. The regular level set theorem makes $F^{-1}(-1)$ a three-dimensional embedded submanifold; intersecting it with the open set $\{x_0>0\}$ preserves that property. Therefore $X$ is the three-dimensional hyperboloid model, and