The Lie group $Sp(2) = \{A\in GL(2,\mathbb{H})\mid A^\dagger A = I \}$ has a variety of nice geometric aspects. One of which is that it is the boundary of the disk bundle $D(V)$ of the rank-1 quaternionic vector bundle $V\to S^7$ associated to $Sp(1) \to Sp(2) \to S^7$. Here $Sp(1) \to Sp(2)$ is the inclusion of the top left entry, $q\mapsto \left(\array{q & 0\\0&1} \right)$.


If one looks at the fibre of this disk bundle over the image of $I$ in $S^7$, then it turns out to be the unit ball in $\mathbb{H}$, which has boundary $Sp(1)$. It thus has the nice extrinsic definition as $D(V)\_{[I]}=\{q\in \mathbb{H} \mid \bar q q \leq 1 \}$. What I'm wondering is if there is such a nice description of all of $D(V)$ as a submanifold-with-boundry of $End(\mathbb{H}^2)$.


One can then consider the framing on $Sp(2)$ arising from this embedding, analogous to how one can consider the framing on spheres arising from their standard embeddings. Is this known explicitly?



Manifold bounded by Sp(2) realisable inside End(H^2)?
The answer to the first question goes as follows. Let $D=[0,1]$, considered as a submonoid of $End(\mathbb{R})$, and include it in $End(\mathbb{H})$, and then include that into $End(\mathbb{H}^2)$, so (the image of) $D$ consists of matrices of the form $\left(\array{ r & 0 \\ 0 & 1}\right)$. Now consider $D(V):=\{ B \in End(\mathbb{H}^2) \mid B^\dagger B \in D \}$. There is a map $End(\mathbb{H}^2) \to \mathbb{H}^2 \hookrightarrow End(\mathbb{H}^2)$ that replaces the first column by zeroes, this restricts to a map $D(V) \to S^7$. In this way, $D(V)$ is the disk bundle of a plane bundle $V$ embedded in $End(\mathbb{H}^2)=\mathbb{H}^2\oplus \mathbb{H}^2$.


We can describe this plane bundle ﻿thusly: consider $S^7 \subset \{0\}\oplus\mathbb{H}^2 \subset \mathbb{H}^2\oplus \mathbb{H}^2$, and for a point $(p,q)\in S^7$ take the fibre $V\_{(p,q)}$ to be $\{(a,b;p,q)\in \mathbb{H}^2\oplus \mathbb{H}^2 \mid \bar a p + \bar b q = 0\}$. The subset consisting of $(0,0;p,q)$ is the image of the zero section, the unit disks in each fibre assemble to give $D(V)$ above, and the unit sphere bundle is $Sp(2)$.


