I'm looking for examples of smooth functions $f: \mathbb{R}^2 \longrightarrow [0,\infty[$ such that the determinant of their Hessian matrix does only vanish on the unit circle, i.e.
$$\det \mathsf{H}_f(x,y)=f_{xx}(x,y)f_{yy}(x,y) - f_{xy}^2(x,y)=0 \Longleftrightarrow x^2+y^2=1.$$
What can be said about the overall shape of such a function?
For $f(x,y)=g(x^2+y^2)$ it follows $$\det \mathsf{H}_f(x,y)=4 g'(x^2 + y^2) (2 (x^2 + y^2) g''(x^2 + y^2) + g'(x^2 + y^2)).$$ Such that this vanishes on a circle iff $$g'(1)=0 \ \ \text{or} \ \ 2g''(1) + g'(1)=0.$$