TITLE: Is every Lie group the automorphism group of a riemannian manifold?
QUESTION [9 upvotes]: Given a finite-dimensional Lie Group $G$, is there always a Riemannian manifold $M$, such that $G$ is the group of isometries of $M$?

REPLY [4 votes]: As Torsten Hĕrculĕ Cärlemän points out, this question has been answered on MathOverflow:
Can every Lie group be realized as the full isometry group of a Riemannian manifold?