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- W2888429256 abstract "We use the octonionic multiplication $cdot$ of $mathbb{S}^{7}$ to associate, to each normal section $eta$ of a submanifold $M$ of $mathbb{S}^{7},$ an octonionic Gauss map $gamma_{eta}:Mrightarrowmathbb{S}^{6}$ given by $gamma_{eta}(x)=x^{-1}cdoteta(x),$ $xin M,$ where is $mathbb{S}^{6}$ the unit sphere of $T_{1}mathbb{S}^{7},$ $1$ the neutral element of $cdot$ in $mathbb{S}^{7}.$ If $eta$ and $nu$ are unit normal sections of $M$ an inner product $langle S_{eta},S_{nu}rangle$ between the second fundamental forms $S_{eta}$ and $S_{nu}$ of $M$ is defined as the trace of the bilinear form [ (X,Y)in TMtimes TMmapstolangle S_{eta}(X),S_{nu}(Y)rangleinmathbb{R}. ] We prove that if $M$ is a codimension $2leq kleq5$ minimal submanifold of $mathbb{S}^{7}$ admitting $k$ orthonormal normal sections $eta_{1}% ,dots,eta_{k}$, parallel in the normal bundle then $S_{eta_{1}}% ,...,S_{eta_{k}}$ are orthogonal if and only if $gamma_{eta_{j}}:Mrightarrowmathbb{S}^{6}$ is harmonic for all $2leq jleq k.$ Considering $mathbb{S}^{m}$ as a totally geodesic submanifold of $mathbb{S}^{7}$, $3leq mleq7,$ we then obtain that if $M^{m-1}$ is an oriented minimal hypersurface of $mathbb{S}^{m}$ and $eta$ is an unit normal vector field to $M$ in $mathbb{S}^{m}$ then $gamma_{eta}$ is a harmonic map. We prove that if $M$ is compact and satisfies the hypothesis of one of the above theorems with $langle S_{eta_{i}},S_{eta_{j}}rangle=0$ if $ineq j,$ $2leq i,jleq k,$ in the first theorem, then the image of any associated octonionic Gauss map is not contained in an open hemisphere of $mathbb{S}^{6}.$" @default.
- W2888429256 created "2018-08-31" @default.
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- W2888429256 date "2018-08-21" @default.
- W2888429256 modified "2023-09-27" @default.
- W2888429256 title "A note on minimal submanifolds of $mathbb{S}^n$ and harmonic octonionic Gauss maps for $3leq nleq7$" @default.
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