This might be a stupid question. The constant rank level set theorem says that if I have a smooth map $f:M\to N$ and a regular point $p\in N$, then if on $U_p$ the rank of the differential is constant, then the preimage of $p$ is a submanifold. The regular level set theorem only demands that the differential with respect to the preimages of $p$ is surjective. For the constant rank level set theorem to imply regular level set theorem, I need to know the surjectivity on the point $p$ implies the constant rank on some neighborhood of $p$. But I think this is not generally true. I'd appreciate any help!
Does constant rank level set theorem imply regular level set theorem
smooth-manifolds
Best Answer
Suppose $q\in M$ is a preimage of $p$ and thtat the map at $q$ has maximal rank. You can then find charts $V\subset M$ and $U\subset N$ such that $f(U)\subset V$ (because $f$ is smooth).
Consider now $g$ to be $f$ in the coordinates given by these charts, say $g = (g_1..., g_n)$. The rank of $f$ is the same as the rank of $g$ and the latter one can be computed in the Jacobian: $Dg = \left(\dfrac{dg_i}{dx_j}\right)$. That the rank is maximal it means that you can find a subdeterminant of maximal size (the biggest of the matrix dimensions) that does not vanish.
Notice that if you were working with a smaller rank, and not the maximal rank, that you can find a subdeterminant of the appropiate size that does not vanish is still true, but not equivalent to your matrix having that given rank since nothing assures you there is not a bigger subdeterminant which also doesnt vanish. That is, the rank might jump up. This of course doesnt occur with maximal rank since no bigger determinant can be found.
Since the determinant of this subsquare matrix is a polynomial in the $\dfrac{dg_i}{dx_j}$, the condition of it not vanishing is an open condition. That is, there is a neighborgood $U_q$ around $q$ such that the rank is constant since it is the maximal possible rank. Varying $q$ on the fiber of $p$, and making a union over all the previous neighborhoods $U_q$, implies that indeed there is an open neighborhood around it such that the rank is constant there and now you can apply the constant rank theorem to conclude the preimage is actually a regular submanifold.