Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers

Wen, Haomin

doi:10.2140/agt.2014.14.407

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Lipschitz minimality of the multiplication maps of unit complex, quaternion and octonion numbers

Haomin Wen

Algebraic & Geometric Topology 14 (2014) 407–420

DOI: 10.2140/agt.2014.14.407

Abstract

We prove that the multiplication maps $S^{n} \times S^{n} \to S^{n}$ ( $n = 1, 3, 7$ ) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range, the unique Lipschitz constant minimizers in their homotopy classes. Other geometrically natural maps, such as projections of Hopf fibrations, have already been shown to be, up to isometries, the unique Lipschitz constant minimizers in their homotopy classes, and it is suspected that this may hold true for all Riemannian submersions of compact homogeneous spaces. Using a counterexample, we also show that being a Riemannian submersion alone without further assumptions (like homogeneity) does not guarantee the map to be the unique Lipschitz constant minimizer in its homotopy class up to isometries, even when the receiving space is just a circle.

Keywords

Lipschitz, minimizer, quaternion, octonion, Clifford algebra

Mathematical Subject Classification 2010

Primary: 53C23

Secondary: 53C30, 55R25, 53C43

References

Publication

Received: 5 May 2013

Revised: 31 July 2013

Accepted: 1 August 2013

Published: 23 January 2014

Authors

Haomin Wen
Department of Mathematics University of Pennsylvania David Rittenhouse Laboratory 209 South 33 Street Philadelphia, PA 19104-6395 USA
http://www.math.upenn.edu/~weh/

Volume 14, issue 1 (2014)