Fermionic Zero Modes in Gauge and Gravity Backgrounds on a Sphere

In this letter we study fermionic zero modes in gauge and gravity backgrounds taking a two dimensional compact manifold $S^2$ as extra dimensions. The result is that there exist massless Dirac fermions which have normalizable zero modes under quite general assumptions about these backgrounds on the bulk. Several special cases of gauge background on the sphere are discussed and some simple fermionic zero modes are obtained.