nLab Eschenburg space

Idea

Eschenburg spaces (also Eschenburg manifold or Eschenburg biquotients) are special 7-manifolds obtained as biquotient spaces of the third special unitary group SU(3) by various free group actions of the first unitary group U(1). A similar construction using a quotient space of different actions results in the related Aloff-Wallach spaces. Both are related to the more well-known lens spaces and are of interest in Riemannian geometry.

Definition

Let $p=(p_1,p_2,p_3)\in\mathbb{Z}^3$ and $q=(q_1,q_2,q_3)\in\mathbb{Z}^3$ be two triples of integers with equal sums, so $p_1+p_2+p_3=q_1+q_2+q_3$. U(1) acts on SU(3) by:

$$z*A =diag(z^{p_1},z^{p_2},z^{p_3})A diag(z^{q_1},z^{q_2},z^{q_3})^{-1}.$$

If the action is free, then the quotient space $E_{p,q}\coloneqq SU(3)/U(1)$ is a smooth 7-manifold, called Eschenburg space.

References

Named after the original discussion in:

• Jost-Hinrich Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66, 469-480 (1982) [doi:10.1007/BF01389224]