\{For the entire collection see MR 96h:53003.\}

Previous Review

\{For the entire collection see MR 90m:53004.\}

Previous Review

Previous Review

Previous Review

A. Gray [ibid. 141 (1969), 465--504; MR 39 #4790; MR errata EA 41] generalized Calabi's results by considering vector cross products on manifolds. He proved that every orientable 6-dimensional submanifold $M\sp 6$ of an arbitrary 8-dimensional pseudo-Riemannian manifold $M'$ possessing a 3-fold vector cross product has an almost complex structure. In particular, if $M'=R\sp 8=O$, every 6-dimensional orientable submanifold of $O$ has a $\text U(3)$-structure.

In the paper under review the author studies the 6-dimensional orientable submanifolds of $O$ by a more direct approach, using the moving frames method. The $\text U(3)$-structure of $M\sp 6$ is constructed along the following lines. Let $u\in S\sp 6$ be a unit imaginary vector of $O$. Let $O\sb u$ be the Hermitian vector space whose underlying real vector space (with inner product) is $O$ and whose complex structure is given by $J\sb u$. It is known that every oriented 6-plane in $O$ is a complex 3-plane in $O\sb u$ for a unique $u\in S\sp 6$. This implies that every oriented 6-dimensional submanifold of $O$ carries an almost Hermitian structure.

The paper starts by giving a brief description of the octonion inner product algebra $O$ and its properties. The group $\text{Spin}(7)$ is defined as the subgroup of $\text{SO}(8)$ generated by the operators $J\sb u$. Then a representation of the Lie algebra $\text{spin}(7)$ by $8\times 8$ complex-valued matrices is given and the structure equations of $\text{Spin}(7)$ are established. These equations, together with the structure equations of the standard $\text{Spin}(7)$-structure of $O$, are extensively used throughout the rest of the paper. The geometry of the Grassmannian $G(2,O) [G(6,O)]$ of oriented 2-planes [oriented 6-planes] in $O$ is investigated in Section 2. Sections 3 and 4 contain the main results.

Section 3 is devoted to the study of the 6-dimensional oriented submanifolds $M\sp 6$ of $O$ endowed with the almost Hermitian structure described above. Using the moving frames method, the second fundamental form is easily determined. With respect to the almost Hermitian structure of $M\sp 6$, II decomposes into three pieces. This allows the author to characterize the immersions $X\colon M\sp 6\rightarrowO$ such that the induced almost Hermitian structure on $M\sp 6$ satisfies some extra condition, e.g., it is complex or symplectic or Kahler, etc. (It must be noted that the present author's terminology employed to denote the special classes of almost Hermitian manifolds is different from that of Calabi and Gray.) Those theorems which generalize Calabi's theorems were proved by Gray using different methods. However, in the present paper, the even more difficult problem concerning the existence of an immersion $X\colon M\sp 6\rightarrowO$ whose induced $\text U(3)$-structure is complex but non-Kahler is examined. It is proved that if $X$ is a complex non-Kahler immersion free of Kahler-umbilics then $M\sp 6$ is foliated by 4-planes in $O$ in a unique way. This foliation is referred to as the asymptotic ruling of $M\sp 6$. The existence and "generality" of various types of complex immersions is investigated using the theory of exterior differential systems.

The final section is devoted to the study of the "complex curves" in $S\sp 6\subseteq\text{Im}\,O$, i.e., of the maps $\varphi\colon M\sp 2\rightarrow S\sp 6$, where $M\sp 2$ is a Riemann surface and $d\varphi$ is complex linear with respect to the almost complex structure on $S\sp 6$ induced by the inclusion in $\text{Im}\,O$. A Frenet formalism is developed; the first, second and third fundamental forms are defined as holomorphic sections of line bundles over $M\sp 2$. In particular, it is shown that the third fundamental form III, analogous to the torsion of a space curve, plays a crucial role. The hypothesis III $\neq 0$ is very restrictive and the author proves that if $M\sp 2=P\sp 1$ then III $\neq 0$ is impossible. On the contrary, he constructs complex curves $\varphi\colon M\sp 2\rightarrow S\sp 6$ with III $=0$ for any Riemann surface $M\sp 2$ such that the ramification divisor of $\varphi$ has arbitrarily large degree.

Previous Review

Previous Review

Return to List

(c) 2002, American Mathematical Society