  In this paper, we define locally convex vector spaces of weighted vector
fields and use them as model spaces for Lie groups of weighted diffeomorphisms
on Riemannian manifolds. We prove an easy condition on the weights that ensures
that these groups contain the compactly supported diffeomorphisms. We finally
show that for the special case where the manifold is the euclidean space, these
Lie groups coincide with the ones constructed in the author's earlier work
["Weighted diffeomorphism groups of Banach spaces and weighted mapping groups".
In: Dissertationes Math. 484 (2012), p. 128. DOI: 10.4064/dm484-0-1].
