A note on skew-Hopf fibrations
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- by Michael E. Gage PDF
- Proc. Amer. Math. Soc. 93 (1985), 145-150 Request permission
Abstract:
Each great circle fibration of the unit $3$-sphere in $4$-space can be identified with a subset of the Grassmann manifold of oriented $2$-planes in $4$-space by associating each great circle fiber with the $2$-plane it lies in. This Grassmann manifold can be identified with the space ${S^2} \times {S^2}$. H. Gluck and F. Warner, in Great circle fibrations of the three sphere, Duke Math. J. 50 (1983), 107-132, have shown that the subsets of this Grassmann manifold which correspond to great circle fibrations can be interpreted as the graphs of distance decreasing maps from ${S^2}$ and ${S^2}$ and that Hopf fibrations correspond to constant maps. This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the $3$-sphere obtained from Hopf fibrations by applying a linear transformation of $4$-space followed by projection of the fibers back to the unit $3$-sphere.References
- E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
- Herman Gluck and Frank W. Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50 (1983), no. 1, 107–132. MR 700132
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 145-150
- MSC: Primary 55R25; Secondary 57R30
- DOI: https://doi.org/10.1090/S0002-9939-1985-0766545-1
- MathSciNet review: 766545