ORIENTED 6-DIMENSIONAL SUBMANIFOLDS IN THE OCTONIANS, III

. In this paper, we classify 6-dimensional almost Hermitian submanifolds in the octonians O according to the classification introduced by A. Gray and L. Hervella. We give new examples of quasi-K/thler and *-Einstein submanifolds in O. Also, we prove that a 6-dimensional weakly ,-Einstein Hermitian submanifold in O is totally geodesic.


INTRODUCTION.
Let (M6, t) be an oriented 6-dimensional submanifold in the 8-dimensional Euclidean space IRs. Throughout this paper, we shall identify IR s with the octonians O (or Cayley algebra) in the natural way. Taking account of the algebraic properties of the octonians, we may observe that M has an almost Hermitian structure. A. Gray and L. Hervella defined the sixteen classes of almost Hermitian manifolds ( [6]). In 3, we classify 6-dimensional almost Hermitian submanifolds in O according to the classification given by A. Gray and L. Hervella. Consequently, the number of different classes (in the sense of A. Gray and L. Hervella) of almost Hermitian submanifolds in O is six. Last of this section, we give examples of quasi-K/iller (or (1,2)-symplectic) submanifolds in O which are not nearly Khler and normal connections are not flat. In 4, we shall investigate the weakly ,-Einstein submanifolds in O ( [13]). We see that any Einstein Khler manifold is necessarily a weakly ,-Einstein manifold. First, we give some examples of ,-Einstein submanifolds in O whose ,-scalar curvature vanish. Also we shall show that a weakly ,-Einstein Hermitian submanifold in is a totally geodesic submanifold (Theorem 4.6). This result is a slight generalization of Chern's result ( [4]) for n 4 in our situation.
In this paper, we adopt the same notational convention as in [1], [10]  , , , , , . O}, wh,'r' \ is the map from SO(S)to itself ch'fin,xl by (9)(c)-9(g-(1)e'] for any ' Then we may observe that k Sp,,(r)"$p(7)-S0(7)is a doul)h, covering map and satisfies the fl,llowing equivariance" 9(u)x g(c) X(g)(u x ,') fo any 9 S1,, (7). where x is the vecov cro.s prodact defin'd by u x'=(Yu-g')/2. Now, we shall txll the structure equations of an olicnted 6-diu('nsional submanlfold n (O, SI)17I It , kiown that considered ab the algebra H,SH where H is the quartenions We put a 1)asis of C E-N, E-jN, Ea-kN, N, E-N, E= iN, E-kN where e =(0,1)HPH, N (1le)/2,2 (1 + -1)/2 e C @ RO and{1, ,, j, k} is the canonical basis of H. We for X ffT,M 6, where , r/ are an orthonormM pair of the normM vector fields and n ((v/'-1,) (for details, see [1], [10]). 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 M be an oriented 6-dimensional manifold and t, ':M--,O be isometric immersions. If there exists g Spin(7) such that t'= g (up to parallel displacement) then J J" where J and d" are the almost complex structures on M 6 induced by the immersions and t', respectively. Also, we can easily see that T'= sanc{f,f,f where T ' is the subbundle of the complexified tangent bundle TM 6 G}C whose fibre is --eigenspace of the almost complex structure d.
Then we have the following structure equations:  for any i,j 1,2,3.
PROOF. By the assumption, we see that M is a quasi-Kthler submanifold in O. From ([10], p 84),it follows that the rank ofBis3or or0. If the rank of Bis 3, then Mistotally un-bilic by Theorem A in [10]. Hence we get case (2). If the rank of B is 1, we see that the normal connection of M is fiat. By Theorem B in [10], ve get case (3). If the rank of B is 0, by  for any x T,M . We note that the induced almost complex structure of M and the almost complex structure of the normal bundle depend only on the conformal structure of the induced metrics. Also, we see that the condition (3.2) is equivalent to the ellipse of curvature of the immersion ry. being a circle. If the ellipse of curvature is a circle, then it is a conformal invariant [7]. On the other hand, since the immersion is a super-minimal immersion, the ellipse of curvature of the immersion is a circle. Also since r,, is a conformal mapping, we see that p the immersion r,,satisfies the condition (3.2). REMARk  REMARK. In general, r, is not constant. 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.   PROOF. By Theorem B in [9], the induced almost Hermitian structure on S(r)x S(r)is not quasi-K/ihler.
REMARK. The complex structure on SSx S which is defined by Calabi-Eckmann is never ,-Einstein [13]. Also the 3-symmetric space Sp(1) x Sp(1) x Sp(1)/Sp(1) S x S is a nearly-K/ilher Einstein, ,-Einstein manifold where Sp(1) is embedded diagonally in Sp(1) x Sir(l) x Sp(1). Hence the induced almost complex structure is different from the above two almost complex structures.
Next, we shall study fundamental relations between ,-scalar curvature and the second fundamental form of M.    Then we have rankC(p) and that if rankC(p)= 1, then there exists a neighborhd U of p such that C *cc and A (*ac + 'ca), where a,c e M xa(C)-valued functions on (V) which are well-defined up to sign.
We are now in a position to prove Theorem 4.6. If rankC 1, by Lemma 4.8, we have