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

Ufa Mathematical Journal. 2018. Vol. 10. No. 2. P. 44–57.

Zhukova N.

We study the groups of conformal transformations of 𝑛-dimensional pseudo-
Riemannian orbifolds (𝒩, 𝑔) as 𝑛 > 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 ∈ {0, 1}∪{3, . . . , 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 > 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.

Keywords: Lie group, orbifold, conformal transformation, conformal pseudo-Riemannian geometry

Publication based on the results of: Жесткие геометрии и группы их автоморфизмов на сингулярных пространствах слоев слоений со связностями Эресмана (2016)