For which $t$ is followins set a differential manifold $$ M_t=\{(x,y,z):x^2+y^2+z^2=1,x^2-y^2+tz^3=1\} $$ I tried to analyze possible determinants of $2\times2$ matrixes from $ \begin{bmatrix}2x & 2y & 2z \\2x & -2y & 3tz^2 \end{bmatrix}$ but couldn't obtain acceptable result.
For what $t$ is $M_t=\{(x,y,z):x^2+y^2+z^2=1,x^2-y^2+tz^3=1\}$ differential manifold?
2
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real-analysis
manifolds
smooth-manifolds
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0The right upper entry of the Jacobian should be $2z$. – 2017-01-03
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0It's ok now. Now I think my method weren't correct. Even if I find all $(x,y,z)$ where relevant determinants are 0, it wouldn't be enough to know exactly what happens with "manifoldity" in these points. Or maybe it would be if that points were excluded from $M_t$ – 2017-01-03
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0I do not have an answer now, but I draw both surfaces for different values of $t$. They do not intersect for small positive values of $t$ whereas they do for large values. The later seems to be a smooth curve. – 2017-01-03
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0Did you draw this by yourself or with help of some software? – 2017-01-04
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0@SekstusEmpiryk I used Scientific work place. – 2017-01-09