Influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds

Abstract

We study the groups of conformal transformations of $n$-dimensional pseudo-Riemannian orbifolds $({\mathcal N},g)$ as $n\geq 3$. We extend the Alekseevskii method for studying conformal transformation groups of Riemannian manifolds to pseudo-Riemannian orbifolds. We show that a conformal pseudo-Riemannian geometry is induced on each stratum of such orbifold. Due to this, for $k\in\{0,1\}\cup\{3,\ldots,n-1\}$, we obtain exact estimates for the dimensions of the conformal transformation groups of $n$-dimensional pseudo-Riemannian orbifolds admitting $k$-dimensional stratum with essential groups of conformal transforms. A key fact in obtaining these estimates is that each connected transformation group of an orbifold preserves every connected component of each its stratum. The influence of stratification of $n$-dimensional pseudo-Riemann orbifold to the similarity transformation group of this orbifold is also studied for $n\geq 2$. We prove that the obtained estimates for the dimension of the complete essential groups of conformal transformations and the similarity transformation groups of $n$-dimensional pseudo-Riemann orbifolds are sharp; this is done by adducing corresponding examples of locally flat pseudo-Riemannian orbifolds.