Indifference curve - Does $dU = 0$ hold in higher dimensions? / Problem of integrability

In two dimensions, we have on an indifference curve that $dU=0$.

Does this apply to indifference objects in higher dimensions?

I was thinking that if $dU > 0$, then one is moving to a higher indifference curve/object

My Macroeconomics professor, in the first week of class, gave a microeconomics review. I asked if $dU = 0$ in higher dimensions. He said he would answer the question next time, and during the next class, he said it may not hold true in higher dimensions because of the problem of integrability.

I asked another economics professor in the department who said that $dU=0$ holds by definition of indifference curves/objects.

What is this problem of integrability, and what does it have to do with indifference curves?

Is it this one?

I’ve taken 2 courses of real analysis, 2 on probability theory and 2 on stochastic calculus (and 4 on statistics), but the mathematics I’ve encountered in economics is only up to basic linear algebra and separable ordinary differential equations (Actually the linear algebra’s mainly just matrices. I hardly recall learning concepts like ‘kernel’ or ‘eigenvalue’ in economics, though I have learned uses for both in finance).