On a non-linear equation related to the geometry of the diffeomorphism group (Q811625)

Let \(M\) be a compact boundaryless Riemannian manifold. We derive the equations on \(M\) which characterize asymptotic vectors on \(\hbox{Diff}_{vol}(M)\). We classify those \(M\)'s whose volume-preserving diffeomorphism groups admit asymptotic vectors which are represented by harmonic vector fields on \(M\). We then show that these harmonic solutions can be used to construct other (typically non-harmonic) solutions.