  Using basic homotopy constructions, we show that isomorphism classes of
string structures on spin bundles are naturally given by certain degree 3
cohomology classes, which we call string classes, on the total space of the
bundle. Using a Hodge isomorphism, we then show that the harmonic
representative of a string class gives rise to a canonical 3-form on the base
space, refining the associated differential character. We explicitly calculate
this 3-form for homogeneous metrics on 3-spheres, and we discuss how the
cohomology theory tmf could potentially encode obstructions to positive Ricci
curvature metrics.
