Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ acts on $(M,g,\omega)$ with corresponding moment map $\mu\colon M\to\mathbb{R}^{\ast}$. The symplectic reduction manfiold (Marsden-Weinstein reduction)

$M_{r} = \mu^{-1}(0)/G$

is a symplectic manifold with symplectic form induced by $\omega$.

My question is, what is the signature of the induced metric on $M_{r}$ by $g$? Is it possible, starting with a metric of indefinite signature, to obtain a Riemannian metric on $M_{r}$?

Thanks.