Quasi-positive curvature on a biquotient of $Sp(3)$

TL;DR

This paper constructs a Riemannian metric with quasi-positive curvature on a specific biquotient of the symplectic group Sp(3), expanding the class of known manifolds with such curvature properties.

Contribution

It introduces a new example of a biquotient of Sp(3) that admits a quasi-positively curved metric, demonstrating a novel geometric construction.

Findings

The biquotient H_1ackslash Sp(3)/H_2 admits a quasi-positively curved metric.

Explicit construction of the metric on the biquotient.

Extension of known examples of manifolds with quasi-positive curvature.

Abstract

Suppose $\phi_3:Sp(1)\rightarrow Sp(2)$ denotes the unique irreducible $4$-dimensional representation of $Sp(1) = SU(2)$ and consider the two subgroups $H_1, H_2\subseteq Sp(3)$ with $H_1 = \{\operatorname{diag}(\phi_3(q_1), q_1): q_1 \in Sp(1)\}$ and $H_2 = \{\operatorname{diag}(\phi_3(q_2),1):q_2\in Sp(1)\}$. We show that the biquotient $H_1\backslash Sp(3)/H_2$ admits a quasi-positively curved Riemannian metric.