Geodesics in Minimal Immersions of $S^3$ into $S^{24}$

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.