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Proceedings of the American Mathematical Society
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 | References | Additional Information

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 Furthermore, the relationships which hold when $ F = {\mathbf{R}},{\mathbf{C}}$ are no longer valid for $ F = {\mathbf{H}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0397750-9
PII: S 0002-9939(1976)0397750-9
Keywords: Axial map, embedding, immersion, projective space, skew map, symmaxial map, tangent bundle
Article copyright: © Copyright 1976 American Mathematical Society



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