A note on totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces

TL;DR

This paper demonstrates that all 7-dimensional Eschenburg spaces can be totally geodesically embedded into infinitely many 13-dimensional Bazaikin spaces, but positive curvature embeddings are not always possible.

Contribution

It establishes the existence of totally geodesic embeddings of Eschenburg spaces into Bazaikin spaces and highlights limitations related to curvature conditions.

Findings

Every 7-dimensional Eschenburg space can be embedded into infinitely many Bazaikin spaces.

Positive curvature embeddings are not always achievable under known constructions.

Examples show curvature constraints prevent some embeddings.

Abstract

In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known construction, it is not always possible to totally geodesically embed a positively curved Eschenburg space into a Bazaikin space with positive curvature.