Totally geodesic embeddings of 7-manifolds in positively curved 13-manifolds

We demonstrate a correspondence between the set of Bazaikin spaces with a subset of Aloff-Wallach and Eschenburg spaces. We show that each Bazaikin space contains at least one and generically 10 totally geodesically embedded Aloff-Wallach or Eschenburg spaces. We use these embeddings to get a sharp upper bound for the pinching of the standard biquotient metrics over the whole family of Bazaikin spaces, and to characterise the curvature properties and regularity of the Bazaikin space in terms of the curvature and regularity of the embedded spaces.