In 1966, Chern asked which compact orientable manifolds carry a Riemannian metric and a vector field which is parallel under this metric. An earlier paper answered this without assuming orientability.

Li Haizhong, Li Zhibo, ON SPACE-LIKE SURFACES WITH PARALLEL MEAN CURVATURE VECTOR OF AN INDEFINITE SPACE FORM, Tsukuba Journal of Mathematics, Vol. 19, No. 1 (June 1995), pp. 141-152 ...