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Hermitian forms and the fibration of spheres


Author: Paul Binding
Journal: Proc. Amer. Math. Soc. 94 (1985), 581-584
MSC: Primary 55R25; Secondary 15A63
DOI: https://doi.org/10.1090/S0002-9939-1985-0792264-1
MathSciNet review: 792264
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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

$\displaystyle ({x_1},{x_2}) \to ({\left\vert {{x_1}} \right\vert^2} - {\left\vert {{x_2}} \right\vert^2},2{x_1}{\bar x_2}):{S^{2n - 1}} \to {S^n}.$

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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0792264-1
Article copyright: © Copyright 1985 American Mathematical Society

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