Generalisation of convexity

Question

Consider the function $f(\theta,\phi) = \sin \left(\theta+\phi/2\right)+\sin \left(\theta-\phi/2\right)$. Defined on the torus $(x,y) \in [-\pi,\pi]^2$ with periodic boundary conditions.

This function has the property that following any downhill (uphill) trajectory always arrive at the global minimum (maximum). This property is analogous to convex functions, however $f$ is not convex.

What property describes this feature of $f$?

Neither of the wiki definitions of pseudo-convex or quasi-convex apply to this function.

Answer 1

I am not sure if this directly applies to your function $f$, but there is a notion for convexity on Riemannian manifolds like the Torus -- it is called geodesic convexity.

Answer 2

If you ignore the presence of stationary points (or if you tweak your downhill trajectory in a way that it can avoid stationary points), then a downhill trajectory always ends in a local minimum. However, all the local minimizers (there is only one) of your function are global minimizers.