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

Hashimoto, Hideya; Ohashi, Misa

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].

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:

[1]	R. L. Bryant, Submanifolds and special structures on the octonians, J. Differential Geom.17(1982), no. 2, 185-232. · Zbl 0526.53055
[2]	T. Fukami and S. Ishihara, Almost Hermitian structure onS6, Tohoku Math. J. (2)7(1955), 151-156. · Zbl 0068.36001
[3]	H. Hashimoto, Characteristic classes of oriented 6-dimensional submanifolds in the octonions, Kodai Math. J.16(1993), no. 1, 65-73. · Zbl 0804.57018
[4]	H. Hashimoto, Oriented 6-dimensional submanifolds in the octonions. III, Internat. J. Math. Math. Sci.18(1995), no. 1, 111-120. · Zbl 0827.53044
[5]	H. Hashimoto, T. Koda, K. Mashimo and K. Sekigawa, Extrinsic homogeneous almost Hermitian 6-dimensional submanifolds in the octonions, Kodai Math. J.30(2007), no. 3, 297-321. · Zbl 1149.53034
[6]	H. Hashimoto and M. Ohashi, Orthogonal almost complex structures of hypersurfaces of purely imaginary octonions, Hokkaido Math. J.39 (2010), no. 3, 351-387. · Zbl 1206.53057
[7]	W. Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity, J. Differential geometry5(1971), 1-38. · Zbl 0219.53045
[8]	T. Takahashi, Homogeneous hypersurfaces in spaces of constant curvature, J. Math. Soc. Japan22(1970), 395-410. · Zbl 0189.22501
[9]	R.