= Counterexample to geodesic one-form injectivity on the round sphere
{title2=$\theta=x_3^2\,dx_1$}
The displayed one-form on the unit round sphere has zero integral on every closed <geodesic>. Indeed, the <antipodal map> pulls it back to its negative, and the second half of an oriented great circle is the antipodal image of the first. Nevertheless $d\theta=2x_3\,dx_3\wedge dx_1$ is nonzero on an open set, so the one-form is not a <closed differential form> and cannot be an <exact differential form>. Thus a curvature-free extension of one-form injectivity fails.
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