Dmitri Alekseevsky, Andreas Kriegl, Mark Losik, Peter W. Michor
Reflection Groups on Riemannian Manifolds
Preprint series: ESI preprints

MSC:

51F15 Reflection groups, reflection geometries, See also {20H10,

53C20 Global Riemannian geometry, including pinching, See Also {31C12, 58B20}

20F55 Coxeter groups, See also {22E40}

22E40 Discrete subgroups of Lie groups, See also {20Hxx, 32Nxx}

Abstract: We investigate discrete groups $G$ of isometries of a complete
connected Riemannian manifold $M$ which are generated by reflections,
in particular those generated by disecting reflections.
We show that these are Coxeter groups,
and that the the orbit space $M/G$ is isometric to a Weyl chamber $C$
which is a Riemannian manifold with corners and certain angle conditions
along intersections of faces. We can also reconstruct the manifold and its action from the
Riemannian chamber and its equipment of istropy group data along the faces.
We also discuss these results from the point of view of Riemannian orbifolds.
Keywords: reflection groups, isometries