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96m:53061 53C40
Hashimoto, Hideya(J-NPIT)
Oriented $6$-dimensional submanifolds in the octonians [octonions].
Almost complex structures (Sofia, 1992), 176--184,
World Sci. Publishing, River Edge, NJ, 1994.

References: 0 Reference Citations: 0 Review Citations: 0

In this paper, the author presents a brief survey of some results concerning six-dimensional submanifolds in the octonions. He starts by reviewing Bryant's construction of an almost complex structure on a six-dimensional oriented submanifold in the eight-dimensional Euclidean space (considered as the Cayley algebra). Next, he proves a number of relations between the characteristic classes of these submanifolds, thereby generalizing a result of Calabi on oriented six-dimensional hypersurfaces in the purely imaginary octonions, and providing necessary conditions for the submanifold to be embedded as a closed subset of the Cayley algebra. Finally, as an application, he determines which six-dimensional, irreducible, compact Riemannian 3-symmetric spaces can be embedded in eight-dimensional Euclidean space, and he computes the characteristic classes of a number of examples of six-dimensional submanifolds of the octonions.

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

Reviewed by Peter Bueken

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92j:53024 53C40 (53C15)
Hashimoto, Hideya(J-NPIT)
Oriented $5$-dimensional submanifolds in the purely imaginary octonians.
J. Korean Math. Soc. 28 (1991), no. 2, 167--182.

References: 0 Reference Citations: 0 Review Citations: 0

Introduction: "In this note, we consider the geometry of the base space ${\rm Im}\,O$ with structure group $G\sb 2$, where ${\rm Im}\,O$ is the purely imaginary octonions and $G\sb 2$ is the automorphism group of the octonions. By the algebraic properties of ${\rm Im}\,O$, we see that any oriented 5-dimensional submanifold $(M\sp 5,\Psi)$ in ${\rm Im}\,O$ has an induced almost contact metric structure, that is, the structure group of the tangent bundle is reducible to ${\rm U}(n)\times 1$. In Section 2, we recall the structure equations of the group $G\sb 2$ established by R. L. Bryant [J. Differential Geom. 17 (1982), no. 2, 185--232; MR 84h:53091]. In Section 3, we write the induced structure equations of $(M\sp 5,\Psi)$ in ${\rm Im}\,O$ derived from these equations. In Section 4, we give the conditions for the induced almost contact structure to be normal. In Section 5, we determine the quasi-Sasakian $(M\sp 5,\Psi)$ and nearly cosymplectic submanifolds $(M\sp 5,\Psi)$ in ${\rm Im}\,O$. These results improve slightly one of K. Kenmotsu [Tohoku Math. J. (2) 23 (1971), 59--65; MR 45 #1083]. In Section 6, we give the relations between the Gauss map and the almost contact metric structure. Lastly we note the condition for an induced almost contact structure to be contact. As an application, we show that there does not exist a contact structure for the submanifolds $(M\sp 5,\Psi)$."

Reviewed by Claude Brada

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91e:53057 53C40 (53C15)
Hashimoto, Hideya(J-NIGATT)
Oriented $6$-dimensional submanifolds in the octonians [octonions]. II.
Geometry of manifolds (Matsumoto, 1988), 71--94,
Perspect. Math., 8,
Academic Press, Boston, MA, 1989.

References: 0 Reference Citations: 0 Review Citations: 1

In Part I [Math. Rep. Toyama Univ. 11 (1988), 1--19; MR 90d:53044], the author defined the notions of nearly Kahler and quasi-Kahler manifolds and proved that Kahler implies nearly Kahler implies quasi-Kahler. In this paper, he shows that an oriented 6-dimensional submanifold immersed in the space of octonions which is nearly Kahler but not Kahler is locally isometric to a standard round 6-sphere. He also characterizes those 6-dimensional submanifolds of the octonions which have flat normal connection and are quasi-Kahler but not nearly Kahler.

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

Reviewed by Thomas E. Cecil

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90d:53044 53C15 (53C40)
Hashimoto, Hideya(J-NIGAT)
Some $6$-dimensional oriented submanifolds in the octonians.
Math. Rep. Toyama Univ. 11 (1988), 1--19.

References: 0 Reference Citations: 0 Review Citations: 1

An almost Hermitian manifold $M$ with almost complex structure $J$ is said to be quasi-Kahler if $(\nabla\sb X J)Y+(\nabla\sb {JX}J)JY=0$ and nearly Kahler if $(\nabla\sb X J)Y+(\nabla \sb YJ)X=0$ for all vector fields $X$ and $Y$ on $M$. A Kahler manifold is nearly Kahler, and a nearly Kahler manifold is quasi-Kahler. In this paper, the author studies these properties on 6-dimensional submanifolds of the space of octonions (Cayley numbers). He proves that if $M\sp 6$ is an oriented quasi-Kahler hypersurface in the unit sphere $S\sp 7$ in the octonions, then it is nearly Kahler and is, in fact, totally umbilic in $S\sp 7$. The octonions can be considered to be $\bold H\times\bold H$, where $\bold H$ is the field of quaternions. The author shows that if the product immersion $M\sp 3\times N\sp 3$ of two hypersurfaces of $H$ is quasi-Kahler in $\bold H\times\bold H$, then it is an open piece of a totally geodesic $\bold R\sp 6$.

Reviewed by Thomas E. Cecil

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87c:81188 81G20 (58G35)
Joshi, G. C.(5-MELB)
Octonian unification and $C,\;P$ and $T$ symmetries.
Lett. Nuovo Cimento (2) 44 (1985), no. 7, 449--454.

References: 0 Reference Citations: 0 Review Citations: 0

Author summary: "A dynamical scheme of quark and lepton family unification is investigated. This scheme is based on a unique nonassociative algebra. This algebraic unification results in three families with $C$-, $P$- and $T$-conserving generation-changing interaction and a fourth family with $P$- and $T$-violating interaction."

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84h:53091 53C55 (22E99 53C40)
Bryant, Robert L.
Submanifolds and special structures on the octonians.
J. Differential Geom. 17 (1982), no. 2, 185--232.

References: 0 Reference Citations: 1 Review Citations: 1

It was proved by E. Calabi [Trans. Amer. Math. Soc. 87 (1958), 407--438; MR 24 #A558] that any oriented submanifold $M\sp 6$ of the hyperplane $\text{Im}\,O$ of the imaginary octonions carries a $\text U(3)$-structure (i.e., an almost Hermitian structure). For example, let $S\sp 6\subseteq\text{Im}\,O$ be the sphere of unit imaginary vectors; then right multiplication by $u\in S\sp 6$ induces a linear transformation $J\sb u\colonO\mapstoO$ which is orthogonal and satisfies $(J\sb u)\sp 2=-I$. The operator $J\sb u$ preserves the 2-plane spanned by 1 and $u$ and therefore preserves its orthogonal 6-plane which may be identified with $T\sb uS\sp 6$. Thus $J\sb u$ induces a complex structure on $T\sb uS\sp 6$ which is compatible with the inner product induced by the inner product of $O$ and $S\sp 6$ has a $\text U(3)$-structure. Calabi has also shown that the second fundamental form II $M\sp 6\rightarrow\text{Im}\,O$ decomposes into two pieces II$\sp {\text{1,1}}$ and II$\sp {\text{0,2}}$ with respect to the $\text U(3)$-structure of $M\sp 6$. The submanifold $M\sp 6$ is complex (i.e., Hermitian) if and only if II$\sp {\text{1,1}}$=0; furthermore its fundamental form is closed if and only if II$\sp {\text{0,2}}$=0. From this it follows that $M\sp 6$ is Kahler if and only if it is a hyperplane (or a union of pieces of a hyperplane). Calabi also constructed nontrivial examples of complex non-Kahler 6-dimensional submanifolds of $\text{Im}\,O$.

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.

Reviewed by Franco Tricerri

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43 #7142 81.22
Penney, R.
Octonians and isospin. (English. Italian, Russian summary)
Nuovo Cimento B (11) 3 1971 95--113.

References: 0 Reference Citations: 0 Review Citations: 0

Author's summary: "The Cayley algebra or algebra of octonions is used to extract the square root of the classical Hamiltonian, in place of the Dirac-Clifford algebra. The resulting wave equation is linear and covariant and possesses a positive definite conserved probability density, and a Lagrangian. The wave function is an octonion, and may be represented as a pair of quaternions. The wave function then possesses an internal degree of freedom which corresponds to multiplication on the right by the Pauli matrices. Lorentz transformation of the wave function is effected by left multiplication, which does not mix up isospin eigenstates. The wave equation seems appropriate for a dynamical description of an isospin doublet. The electromagnetic properties of the objects described by the wave function are those of a Dirac particle with the usual magnetic moment."

Reviewed by T. Sasakawa

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