The Bochner–Lichnerowicz–Weitzenböck formula can be written the following way $$ \Delta \phi-\nabla^*\nabla \phi= R(\phi),$$ here $\phi$ is a section in a Dirac bundle and $R$ the something which can be expressed in terms of its curvature.
One can get something nice applying this formula for different bundles associated to a given Riemannian manifols if $\langle R(\phi),\phi\rangle \ge 0$ for any $\phi$.
For the right choice of bundle, one gets $R$ to be Ricci curvature or scalar curvature and yet number of less popular curvatures. Each of these conditions corresponds to a convex $O(n)$-invariant cone of curvature operators, let us call it Weitzenböck cone.
Question. Is it well understood which cones are Weitzenböck and which are not?
For example assume I have a specific cone (say the cone $sec\ge 0$) is there someone who can tell that it is not a Weitzenböck cone.