Constant rank theorem: intuition?

geometry differential-geometry intuition

Solution 1:

Shouldn't the rank be constant on a neighborhood of that point to make this conclusion? It is sufficient to check at that point if the rank is maximal, but otherwise the rank can "jump" up suddenly, meaning you would need more coordinates. It can't jump down suddenly, because the determinant function is smooth. It takes some time for a nonzero continuous function to become zero, but a function that is zero can "instantly" become non-zero.

Otherwise your intuition seems more or less correct. Sometimes a useful term is locally: "The constant rank theorem says that a smooth function with locally constant rank is locally a linear map of that same rank, up to diffeomorphism."

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