Finding a $1$-form on a hyperboloid of one sheet which is closed but not exact

Question

I want to find a closed 1-form $w$ on $$M=\{(x,y,z):x^2-y^2-z^2=-1\}\subset \mathbb R^3$$ which is not exact. I think that $$\frac{x\,\mathrm{d}x+y\,\mathrm{d}y+z\,\mathrm{d}z}{(x^2+y^2+z^2)^{3/2}}$$ is a such one, but how can I prove that?

Answer 1

This one is exact, unfortunately: it is just $$\mathrm d\biggl(\frac{-1}{(x^2+y^2+z^2)^{1/2}}\biggr).$$

Answer 2

First, a little bit of algebraic-topological intuition. Your set $M$ is a hyperboloid of one sheet, which means it deformation retracts to a circle. A closed, nonexact one-form is a nontrivial cohomology class: It should tell you that a loop around the "waist" of the hyperboloid is not contractible.

So look at the circle. Do you know a closed, nonexact one-form on $S^1 = \{x^2 + y^2 = 1\ |\ (x,y)\in\Bbb{R}^2\}$? Can you adapt that inspiration to construct a closed, nonexact one-form on $M$?

Answer 3

Hint A standard example of a $1$-form that is closed but not exact is the form $$\frac{-z \,dy + y \,dz}{y^2 + z^2}$$ on the punctured plane $\Bbb R^2 - \{0\}$ (with standard coordinates suggestively named); suggestively (but not quite precisely) this is often denoted $d \theta$, where $\theta$ is an angular polar coordinate.

Remark Incidentally, if one raises the sole index of this form using the Euclidean metric, the result is the vortex vector field, $\frac{1}{y^2 + z^2}(-z \partial_y + y \partial_z)$.