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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Axial maps with further structure

Author: A. J. Berrick
Journal: Proc. Amer. Math. Soc. 54 (1976), 413-416
MathSciNet review: 0397750
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Abstract: For $F = {\mathbf {R}},{\mathbf {C}}$ or ${\mathbf {H}}$ an $F$-axial map is defined to be an axial map ${\mathbf {R}}{P^m} \times {\mathbf {R}}{P^m} \to {\mathbf {R}}{P^{m + k}}$ equivariant with respect to diagonal and trivial ${F^{\ast }}$-actions. Analogously to the real case, it is shown that ${\mathbf {C}}$-axial maps correspond to immersions of ${\mathbf {C}}{P^n}$ in ${{\mathbf {R}}^{2n + k}}$ while (for $F = {\mathbf {R}}$ and for $F = {\mathbf {C}}$, $k$ odd) embeddings induce $F$-symmaxial maps. Examples are thereby given of symmaxial maps not induced by embeddings of ${\mathbf {R}}{P^n}$, and of ${\mathbf {R}}$-axial maps which are not ${\mathbf {C}}$-axial. Furthermore, the relationships which hold when $F = {\mathbf {R}},{\mathbf {C}}$ are no longer valid for $F = {\mathbf {H}}$.

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Keywords: Axial map, embedding, immersion, projective space, skew map, symmaxial map, tangent bundle
Article copyright: © Copyright 1976 American Mathematical Society