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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 Furthermore, the relationships which hold when $ F = {\mathbf{R}},{\mathbf{C}}$ are no longer valid for $ F = {\mathbf{H}}$.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