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.



Symplectic reduction: from indefinite signature to Riemannian signature
Just take $R^4=C^2$ with complex 
coordinates $z\_1=x\_1+ i y\_1, z\_2= x\_2 + i y\_2$ and the flat  

metric $-dz\_1 d\bar z\_1 + dz\_2 d \bar z\_2$. It has signature (2,2). As the group of isometries take the group of $x\_1 $-translations, it preserves the canonic symplectic form $dx\_1\wedge dy\_1 + dx\_2 \wedge dy\_2$, and actually is generated by the function $y\_1$. The symplectic reduction gives you a positively definite metric $dz\_2d\bar z\_2$. 


