Let $\mathbb H$ be algebra of quaternions and let $S$ be the group of unit quaternions. What does it mean for a point to be tangent to $\mathcal S$?

Question

Let $\mathbb H$ be the algebra of quaternions and let $\mathcal{S} \subset \mathbb H$ be the group of unit quaternions.

Show that if $p \in \mathbb H$ is imaginary, then $qp$ is tangent to $\mathcal S$ at each $q \in \mathcal S$.

Definition: If $S$ is an immersed submanifold of $M$, and $p \in S$, a vector field $X: M \to TM$ is tangent to $S$ at $p$ if $X(p) \in T_p S \subset T_p M$.

What does it mean for a point, like $qp$, to be tangent to $\mathcal S$?

Answer 1

Since $\mathbb{H}$ is a vector space, the tangent space at each point of $\mathbb{H}$ can be canonically identified with $\mathbb{H}$ itself. So, when it says $qp$ is tangent to $\mathcal{S}$ at $q$, this means that $qp$, considered as a tangent vector to $\mathbb{H}$ at the point $q$, is in the tangent space of the submanifold $\mathcal{S}$.