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June 1990 Geodesics in Minimal Immersions of $S^3$ into $S^{24}$
Yosio MUTŌ
Tokyo J. Math. 13(1): 221-234 (June 1990). DOI: 10.3836/tjm/1270133016

Abstract

In the present paper we consider geodesics which are obtained as images of great circles of $S^3(1)$ induced by an isometric minimal immersion $f:S^3(1)\to S^{24}(r)$, $r^2=1/8$ namely, geodesics of $f(S^3(1))$. $S^{24}(r)$ being regarded as a hypersphere of $\mathbf{R}^{25}$, we can consider such geodesics as curves in $\mathbf{R}^{25}$ with curvatures $k_1,k_2,k_3$. It is found that these are constants which depend on the choice of the geodesic except the case where $f$ is a standard minimal immersion [8]. Equations satisfied by $k_1,k_2,k_3$ and the necessary and sufficient condition for an isometric minimal immersion to have a geodesic which is a circle are obtained.

Though we concentrate our topic upon the case $S^3(1)\to S^{24}(1)$, in the beginning part of the paper some properties of minimal immersions of spheres into spheres in general are recollected with some additional results.

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Yosio MUTŌ. "Geodesics in Minimal Immersions of $S^3$ into $S^{24}$." Tokyo J. Math. 13 (1) 221 - 234, June 1990. https://doi.org/10.3836/tjm/1270133016

Information

Published: June 1990
First available in Project Euclid: 1 April 2010

zbMATH: 0723.53037
MathSciNet: MR1059026
Digital Object Identifier: 10.3836/tjm/1270133016

Rights: Copyright © 1990 Publication Committee for the Tokyo Journal of Mathematics

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Vol.13 • No. 1 • June 1990
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