A note on the existence of nontrivial zero modes on Riemannian manifolds

We prove a necessary criterion for the (non-)existence of nontrivial solutions to the Dirac equation $D\psi =iA{\cdot }_{Cl}\psi$ on Riemannian manifolds that are either closed or of bounded geometry. This generalizes a result of Rupert Frank and Michael Loss on ${\mathbb{R}}^n$ where the criterion relates the $L^n$-norm of $A$ to the Sobolev constant on ${\mathbb{R}}^n$. On Riemannian manifolds the role of the Sobolev constant will be replaced by the Yamabe invariant. If $n$ is odd, we show that our criterion is sharp on ${\mathbb{S}}^n$.