I came across an interesting article by Montesinos (J. Geom. Phys. 2 (1985), no. 2, 145–153.). In it, he finds that spin structures (as lifts of $SO(4)$) are not compatible with all Riemannian metrics ...
One of the remarkable properties of the Kerr black hole geometry is that scalar field equations separate and are exactly solvable (reducible to quadrature), even though naively it does not have enough ...