Isometric actions on pseudo-Riemannian nilmanifolds

TL;DR

This paper investigates the structure of isometry groups of pseudo-Riemannian 2-step nilmanifolds, highlighting differences from Riemannian cases and identifying conditions for automorphism subgroup equality.

Contribution

It provides new insights into the isometry group actions on pseudo-Riemannian nilmanifolds and characterizes when certain subgroups coincide, especially for pseudo-H-type groups.

Findings

Nilradical action need not be transitive

Conditions for isometry subgroup equality are identified

Differences from Riemannian case are demonstrated

Abstract

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation; for instance the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with left-invariant pseudo-Riemannian metric we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-$H$-type groups.