Isometry group of a lorentzian metric which preserves a Riemannian metric I'm wondering if it is possible for the isometry group of a Lorentzian metric to preserve a Riemannian metric. So if $(M,g)$ is a Lorentz manifold and $Iso(M,g)$ its isometry group, I ask myself if there are situations where $Iso(M,g)$ acts by isometries for a Riemannian metric $h$ on $M$.
If we assume $M$ to be compact, then it may happen that for a Lorentz metric $g$ on $M$ the isometry group $Iso(M,g)$ is noncompact. So in this case I think it is not possible for $Iso(M,g)$ to preserve a Riemannian metric on $M$, because the isometry group of a compact Riemannian manifold is compact. But what about Lorentz metrics on $M$ which have compact isometry groups. Can they always preserve a Riemannian metric on $M$? Or is this a very special (and  restrictive) thing?
 A: I do not have a complete answer to this question, but I believe that I have an example which shows that the Isometry group of a Lorentzian manifold $(M, g_{L})$ can act by isometries on a corresponding Riemannian structure $(M, g_{R})$ and vice-a-versa.  (For simplicity I am only considering the connected component of the isometry group which contains the identity.)
Consider $\mathbb{R}^{3}$ with coordinates $(x, y, z)$ and consider the frame field
\begin{align*}
 e_{1} &= \frac{\partial}{\partial x} - \frac{y}{2} \frac{\partial}{\partial z},\\
 e_{2} &= \frac{\partial}{\partial y} + \frac{x}{2} \frac{\partial}{\partial z}, \textrm{ and }\\
e_{3} &= \frac{\partial}{\partial z}. 
\end{align*}
The corresponding dual co-frame field is 
\begin{align*}
\omega^{1} &= \mathrm{d}x,\\ 
\omega^{2} &= \mathrm{d}y,\\ 
\omega^{3} &= \mathrm{d}z + \frac{1}{2}\left(y \mathrm{dx} - x \mathrm{d}y\right).
\end{align*}
Now consider the Lorentzian metric defined by
$$ g_{L} = \omega^{1} \otimes \omega^{1} + \omega^{2} \otimes \omega^{2} - \omega^{3} \otimes \omega^{3}$$
and the Riemannian metric defined by 
$$g_{R} = \omega^{1} \otimes \omega^{1} + \omega^{2} \otimes \omega^{2} + \omega^{3} \otimes \omega^{3}.$$
The connected component of the isometry group  containing the identity for the metrics $g_{L}$ and $g_{R}$ is 
                $$
    Isom(M, g) = \left\{ A \in GL_{4}\left(\mathbb{R}\right) \Big \vert 
    \begin{pmatrix} 
    \cos \theta & -\sin \theta & 0 & a\\
    \sin \theta & \cos \theta & 0 & b\\
    \frac{1}{2}\left(a \sin \theta - b \cos \theta\right) & \frac{1}{2}\left( a \cos \theta + b\sin \theta \right) & 1 & c\\
    0 & 0&0 &1\\
    \end{pmatrix}, 
    a, b, c, \theta \in \mathbb{R}
    \right
    \}.
    $$
Identifying a point $\mathbf{x} = (x, y, z)$ in $\mathbb{R}^{3}$ with $\mathbf{x} =\left( x, y, z, 1\right)$ in $\mathbb{R}^{4}$, then the action of $Isom(M, g)$ on $\mathbb{R}^{3}$ is  given by $A\cdot\mathbf{x}^{t} = \overline{\mathbf{x}}^{t} = \left( \overline{x}, \overline{y}, \overline{z}, 1 \right)$, where
                    \begin{align}
      \overline{x} &= x\cos \theta - y \sin \theta + a\\
      \overline{y} &= x \sin \theta + y \cos \theta + b\\
      \overline{z} &= z + c + \frac{1}{2}\left(a \sin \theta - b \cos \theta\right)x + \frac{1}{2}\left( a \cos \theta + b \sin \theta\right) y.
     \end{align}
Ignoring the $z$-component you have your standard Euclidean isometries on the $xy$-plane.  The key to the example is that the one-form $\omega^{3}$ is invariant under the indicated action.  This allows one to toggle the sign on $\omega^{3} \otimes \omega^{3}$ to obtain either the Lorentzian metric $g_{L}$ or the Riemannian metric $g_{R}$.
(As an aside:  The indicated example comes from three-dimensional Heisenberg group $H^3$ with group structure $(a, b, c)\star(x, y, z) = \left(a + x, b + y, c + z +\frac{1}{2}(ay - bx)\right)$.  The frame field $e_{1}, e_{2}$, and $e_{3}$ is left invariant, as are the corresponding metrics.  The connected-component of the identity of isometry group has the structure of a semi-direct product $H^3 \ltimes SO(2)$.  For details on the Riemannian case see Differential geometry of curves and surfaces in 3-dimensional homogeneous spaces Part I  by Inoguchi et al.)
