Geodesic orbit and naturally reductive nilmanifolds associated with graphs

TL;DR

This paper characterizes when Riemannian nilmanifolds associated with graphs are geodesic orbit and naturally reductive, linking these properties to the structure of the defining graph and specific metrics.

Contribution

It provides a complete characterization of geodesic orbit and naturally reductive nilmanifolds associated with graphs, based on the graph's structure and inner product conditions.

Findings

Nilmanifold is geodesic orbit iff it is naturally reductive.

Such nilmanifolds correspond to graphs that are disjoint unions of complete graphs.

The metric is generated by a naturally defined inner product.

Abstract

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.