A note on skew-Hopf fibrations
Author: Michael E. Gage
Journal: Proc. Amer. Math. Soc. 93 (1985), 145-150
MSC: Primary 55R25; Secondary 57R30
MathSciNet review: 766545
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Abstract: Each great circle fibration of the unit -sphere in -space can be identified with a subset of the Grassmann manifold of oriented -planes in -space by associating each great circle fiber with the -plane it lies in. This Grassmann manifold can be identified with the space . 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 and and that Hopf fibrations correspond to constant maps.
This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the -sphere obtained from Hopf fibrations by applying a linear transformation of -space followed by projection of the fibers back to the unit -sphere.
Keywords: Hopf fibration, great circle fibration, Grassmann manifold
Article copyright: © Copyright 1985 American Mathematical Society