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**On the automorphism groups of isoparametric hypersurfaces of \(S^7\).**
*(English)*
Zbl 1455.53084

Shoda, Toshihiro (ed.) et al., Differential geometry and Tanaka theory. Differential system and hypersurface theory. Proceedings of the international conference, RIMS, Kyoto University, Japan, January, 24–28, 2011. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 82, 75-85 (2019).

The curvature of a hypersurface \(M\) in the ambient manifold is described by the second fundamental form. Its eigenvalues form are the principal curvature functions. A hypersurface in a space of constant curvature is called isoparametric if all these functions are constant.

In this article, isoparametric hypersurfaces \(M^6\) in a seven-dimensional sphere are considered. Such hypersurfaces may be equipped with the induced orthogonal almost complex structures, which are obtained by the multiplication of the octonions. There are eight types of such hypersurfaces.

The groups of automorphisms of the induced orthogonal almost complex structures of such \(M^6\) are calculated here. More accurately, it is done for five types of \(M^6\), for which there are one, two or three principal curvatures. These hypersurfaces \(M^6\) are \(S^6\), \(S^5\times S^1\), \(S^2 \times S^4\), \(S^3 \times S^3\), \(\operatorname{SU}(3)/T^2\). For the remaining three cases there are four (twice) or six principal curvatures.

For the entire collection see [Zbl 1437.53042].

In this article, isoparametric hypersurfaces \(M^6\) in a seven-dimensional sphere are considered. Such hypersurfaces may be equipped with the induced orthogonal almost complex structures, which are obtained by the multiplication of the octonions. There are eight types of such hypersurfaces.

The groups of automorphisms of the induced orthogonal almost complex structures of such \(M^6\) are calculated here. More accurately, it is done for five types of \(M^6\), for which there are one, two or three principal curvatures. These hypersurfaces \(M^6\) are \(S^6\), \(S^5\times S^1\), \(S^2 \times S^4\), \(S^3 \times S^3\), \(\operatorname{SU}(3)/T^2\). For the remaining three cases there are four (twice) or six principal curvatures.

For the entire collection see [Zbl 1437.53042].

Reviewer: V. V. Gorbatsevich (Moskva)

### MSC:

53C40 | Global submanifolds |

53C30 | Differential geometry of homogeneous manifolds |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

### Keywords:

isoparametric hypersurfaces; automorphism groups; octonions; almost complex structure; Spin(7)-congruent### References:

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