Oriented 6-dimensional submanifolds in the octonians, III

Hideya Hashimoto

doi:10.1155/s0161171295000159

Abstract

In this paper, we classify 6-dimensional almost Hermitian submanifolds in the octonians𝕆according to the classification introduced by A. Gray and L. Hervella. We give new examples of quasi-Käthler and∗-Einstein submanifolds in𝕆. Also, we prove that a 6-dimensional weakly∗-Einstein Hermitian submanifold in𝕆is totally geodesic.

Highlights

Let (M6, t) be an oriented 6-dimensional submanifold in the 8-dimensional Euclidean space s.IR Throughout this paper, we shall identify IRs with the octonians O in the natural way
By making u,e of the properties of Spn(7),ve may observe that this almost complex structure is an invariant of Spin(7) in the following sense; let
]rst, we prove the following: PROPOSITION 3.3

Summary

INTRODUCTION

Let (M6, t) be an oriented 6-dimensional submanifold in the 8-dimensional Euclidean space s.IR Throughout this paper, we shall identify IRs with the octonians O (or Cayley algebra) in the natural way. By making u,e of the properties of Spn(7), ,ve may observe that this almost complex structure is an invariant of Spin(7) in the following sense; let. Let M6=(M6, j,g) be a 6-dimensional almost Hermitian submanifold in 13. There are many examples of 6-dimensional oriented Hermitian submanifolds in O which are not K/ihler ([1], [3]). We shall define two classes of almost Hermitian manifolds. We shall give examples of quasi-KXhler submanifolds in O whose normal connections are not flat. -we shall study weakly ,-Einstein submanifolds in O and give some examples. T. Koda [12] proved that (CP#--:, J, 9) is a compact Einstein, weakly ,-Einstein Hermitian surface whose ,-scalar curvature is a non-constant positive function, where 9 is the Berald-Bergery’s metric

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