Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be *string*, if the classifying map of its bundle of orthonormal frames $M \to BO(n)$ lifts to a map $M \to BO(n)<8> = BString(n)$, which is the case if and only if the class $\frac{p_1}{2} \in H^4(M, \mathbb{Z})$ vanishes. There are a lot of models that yield geometric realizations of $String(n)$ either as a topological group (see Stolz-Teichner), infinite-dimensional Lie group (see Nikolaus-Sachse-Wockel) or a 2-group (see Schommer-Pries).

What are enlightening examples of string manifolds? What are non-examples? When do you have a geometric interpretation of the obstruction class?

So far, I am aware of the list given at the end of Douglas-Henriques-Hill. What else is out there?

alwaysa geometric interpretation of the obstruction class, namely as the Chern-Simons 2-gerbe associated to the frame bundle of M and the basic gerbe over Spin. $\endgroup$