Manifold structure and metric on quotient space

If one has a certain (semi)Riemannian manifold $(M,g)$ with a Killing vector field $X$, then the flow $phi$ of $X$, forms a one-parameter isometry group $G$ on $M$. Then one can define an equivalence relation $sim$ between points in $M$ by $psim p’$ iff $p’=h(p)$ for some $hin G$.

I want to know whether the resulting quotient space $M/sim$ is a manifold and if so is there a canonical way of defining a metric on $M/sim$ using the original metric $g$ on $M$?