Hermitian forms and the fibration of spheres

Abstract:

We identify the real $(2n - 1)$-dimensional sphere ${S^{2n - 1}}$ with the unit sphere of ${{\mathbf {F}}^2}$, where ${\mathbf {F}} = {\text {reals}}$, complexes or quaternions and $n = 1,2$ or 4, respectively. It is shown how any Hermitian form over ${{\mathbf {F}}^2}$, restricted to ${S^{2n - 1}}$, is related to the (double covering for $n = 1$, Hopf for $n = 2,4$) fibration \[ ({x_1},{x_2}) \to ({\left | {{x_1}} \right |^2} - {\left | {{x_2}} \right |^2},2{x_1}{\bar x_2}):{S^{2n - 1}} \to {S^n}.\] Convexity of the joint range of several Hermitian forms over the unit sphere of an arbitrary normed vector space $V$ over ${\mathbf {F}}$, with $\dim V > 2$, is deduced as a corollary.