Can every Lie group be realized as the full isometry group of a Riemannian manifold? Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$?
For example if I try to do this for a connected $G$, then I often get a bigger group as the full isometry group, which includes e.g. the orientation reversing isometries. (Maybe one has to take a non--orientable  space for that?)
Even if I try to realize $\mathbb R$ as a full isometry group, I fail. (One could take the full isometry group of $\mathbb R$ with the standard metric, which is given by $\mathbb R \rtimes \mathbb Z_2$ and divide out the $\mathbb Z_2$ action. But this leads to a fixpoint and the quotient is therefore not a manifold any more.)
There is an article of J. de Groot1 which proves that every abstract group can be realized as an isometry group of some metric space, but it is not clear to me, if this is true in the category of Lie groups and Riemannian manifolds.
1de Groot, J. "Groups represented by homeomorphism groups."
Math. Ann. 138 (1959) 80–102.
MR119193
doi:10.1007/BF01369667</a
 A: The article of de Groot is the one cited here: What kind group can be realized as a Isometry group of some space?
That every compact group is the full isometry group of a compact Riemannian manifold is shown in:
Saerens, Rita; Zame, William R., The isometry groups of manifolds and the automorphism groups of domains, Trans. Am. Math. Soc. 301, 413-429 (1987). ZBL0621.32025. (such an isometry group must be compact  priori).
Winkelmann, Jörg, Realizing connected Lie groups as automorphism groups of complex manifolds, Comment. Math. Helv. 79, No. 2, 285-299 (2004). ZBL1056.32022. shows that every connected real lie group is the full automorphism group of a complete, hyperbolic (in the sense of Kobayashi) complex manifold.
A: The group $\mathbb R$ can be realized as full isometry group of $(\mathbb R\times\mathbb S^1 ,g)$.
Choose a generic periodic one parameter family of quadratic forms $h(t)$ on $\mathbb R^2$.
Consider metric $g(x,y)=h(y)$ on $\mathbb R\times\mathbb S^1$.
Why: Note that each fiber $\mathbb R\times u$ maps to it-self.
Note that orthogonal fibration $\mathcal{F}$ is preserved.
Go along $\mathcal{F}$ once around $\mathbb S^1$.
Since $h$ is generic you will not get to the same point.
Therefore each isometry preserves the orientaion of $\mathbb R$-fibers.
This idea seems to work in general. Consider metric $g$ on $G\times \mathbb T^2$ which is invariant w.r.t. left $G$-translations and such that $g(e,t)=h(t)$ is a generic family  of quadratic forms on $T_{(e,t)}$; here $t\in \mathbb T^2$.
Why: This way you get a holonomy map from $G\to G$ for any loop in $\mathbb T^2$.
For generic $h(t)$  you may assume that there is no automorphism of $G$ which preserve this holonomy.
A: Maybe this is the same as the idea of Anton, but I thought I post it anyway for its visualization.
I think something like this will have full isometry group $=\Bbb R$:
a periodically winding "rope" with a generic surface structure.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/9Vv5n.png
In formulas, if $h:\Bbb S^1\to\Bbb R$ is generic, then above figure can be given by a parametric description
$$(\theta,z)\quad\mapsto\quad (1+h(\theta+\alpha z))\begin{pmatrix}\cos(\theta)\\\sin(\theta)\\0\end{pmatrix} + \begin{pmatrix} 0\\0\\z \end{pmatrix}$$
with some parameter $\alpha$.
