About pseudohermitean submanifolds in octonion space.
by
N. M. Kuzub
Coauthors: P.Ja. Grushko

Poster

1. Algebra. Let V be 7-dimensional vector space with anticommutative multiplication [x, y] such that the group of automorphisms of [, ] is exceptional non-compact Lie group G₂ⁿ (normal form of complex Lie group G₂^c), [1]. We call [, ] vector product.

There is an unique scalar product <x, y> on V such that <[x, y], y> = 0 and [x, [x, y]]=<x, y>x-x² y, where x, y in V, which equivalent to the following identity [x, [y, z]]+[y, [x, z]] = <x, z>y+<y, z>x-2<x, y>z where x, y, z in V, [2].

Put \Omega(x, y, z)=<([x, y], x>. Then \Omega is unique up to scalar factor form \Omega in /\ ³ R⁷ such that G₂ⁿ={A in GL(7)/A\Omega = \Omega}.

There exists orthogonal basis {e₁, ..., e₇} of V such that e₁²=e₂²=e₃²=1, e₄²=e₅²=e₆²=e₇²=-1 and
[e₁, e₂]=e₃, [e₁, e₃]=-e₂, [e₁, e₄]=e₅, [e₁, e₅]=-e₄, [e₁, e₆]=-e₇, [e₁, e₇]=e₆, [e₂, e₃]=e₁, [e₂, e₄]=e₆, [e₂, e₅]=e₇, [e₂, e₆]=-e₄, [e₂, e₇]=-e₅, [e₃, e₄]=e₇, [e₃, e₅]=-e₆, [e₃, e₆]=e₅, [e₃, e₇]=-e₄, [e₄, e₅]=e₁, [e₄, e₆]=e₂, [e₄, e₇]=e₃, [e₅, e₆]=-e₃, [e₅, e₇]=e₂, [e₆, e₇]=-e₁.

Definition 1. We call the basis described above octonion frame.

Definition 2. A set of vectors V={v₁, v₂, v₃} is said to be a semi-frame if it generates an (unique) octonion frame E={e₁, ..., e₇} such that V subset E. Thus, the span Rv₁\oplusRv₂\oplusRv₃ is not a subalgebra in V.

2. Surfaces. Let S subset V be an oriented submanifold such that dim  S=6. Denote T_x tangent space at arbitrary point x in S. Let W={w in V/<w, z> = 0    for all  z in T_x} orthogonal (one-dimensional) complement. There are tree cases.
1). <W\{0}, W\{0}> > 0. Then there is unique normal vector N consistent with the orientation such that N²=1. The multiplication Jz=[N, z], z in T_x defines almost complex structure on S. Put \Psi(y, z)=\Omega(y, z, N)=<Jy, z>. Then \Psi in /\ ² T_x^* and H(y, z)=<y, z>+i\Psi(y, z) is pseudohermitean form.
2). <W\{0}, W\{0}> < 0. Then there is unique N in W consistent with the orientation such that N²=-1 and multiplication P(z)=[N, z], z in T_x induces endomorphism P of T_x such that P²=1, T_x=P₊\oplusP_- where P₊={z in T_x/Pz=z}, P_-={z in T_x/Pz=-z}, dim   P₊=dim   P_-=3.
3). <W\{0}, W\{0}> = 0. Then W subset V is isotropic and not normal.

In this paper we deal with the first case only.

We can adjoint to each point x in S an octonion frame {e₁, ..., e₇} such that e₁=N, while {e₂, ..., e₂} is a basis of T_x. To obtain a canonical frame we need to fix two vectors e_i, e_j such that {e₁, e_i, e_j} is a semi-frame.

Since dN induces symmetric bilinear form C(y, z) on T_x we have a symmetric endomorphism A such that C(y, z)=<Ay, x>, y, z in T_x. We suppose that there are no multiple eigenvalues and eigenvectors are not isotropic. Since the signature of <, >|_Tx is (2, 4) there exists an eigenvector v in T_x such that Av=\lambdav, v² > 0. Put e₂=v/|v|. Then e₃=[N, e₂]=Je₂ is also fixed. Further, the signature of orthogonal complement W'={y in T_x/<y, e_i> = 0, i=2, 3} is (0, 4). Thus, there exists vector v' in W' such that Av'=\muv'  mod   (e₂, e₃), v'² < 0. We put e₄=v'/\surd{-v'²}. Then e₁, e₂, e₄ is semi-frame. According to our fixations we have c_i2=0, i=3, ..., 7, c_4j=0, j=5, 6, 7 where c_ij=C(e_i, e_j), \omega_i⁷=c_ij\omega^j.

The derivation formulas look as follows (i, j, \alpha = 2, ..., 7)
dr=\omegaⁱe_i, de_i = \omega_i^je_j+\omega_i¹N=G^j_i\alpha\omega^\alphae_j+b_ij\omega^jN, dN=\omega₁^je_j=c_ij\omegaⁱe_j.

Proposition 1. The Nijenhuis tensor can be calculated in terms of G_ij^k.

Proposition 2. The coefficients c_ij, G_ij^k can be characterized in terms of normal sections defined by pair (N, e_i), i=2, ..., 7 and 2-surfaces in 3-dimensional Euclidean or pseudoeuclidean spaces which are sections defined by triples (N, e_i, e_j).

3. Immersions. Let U subset R⁶ be a domain and f:U --> V an immersion such that f(U)=S. Then df induces differential forms \psi, h on U such that

h(du, du', du'')=\Omega(df(du), df(du'), df(du''))\labeleq:h

(\theequation)

\psi(du, du')=\Psi(df(du), df(du'))\labeleq:psi

(\theequation)

Definition 3. An immersion f:U --> V is said to be regular if (\frac\partialf\partialu₁)² > 0 and (p²q-<p, q>p-\Psi(p, q) p)² < 0 where p=\frac\partialf\partialu₁, q=\frac\partialf\partialu₂.

Proposition 3. For each x in S there is a regular parametrization of S in a small neighborhood of x.

We can construct and calculate in terms of \frac\partialf\partialu_i an octonion (non-canonical!) frame {e'₁, ..., e'₇} associated with immersion f such that e'₁=e₁=N, e₂=k₁\frac\partialf\partialu₁, k₁ > 0, \frac\partialf\partialu₂ = k₂e₄   mod   (e₂, e₃) where k₂ > 0.

The derivation formulas look as follows

df=fi dui, dfi=\Gammai\alphajfjdu\alpha+bijdujN, dN=c'ijduifj \labeleq:dn

(\theequation)

Here f_i=\frac\partialf\partialuⁱ. From , , we get the following equations

\frac\partialhijk\partialum = \Gammaim\alphah\alphajk+\Gammajm\alphahi\alphak+\Gammakm\alphahij\alpha+bim\psijk+bjm\psiki+bkm\psiij\labeleq:first

(\theequation)

\frac\partial\psiij\partialum=\Gammaik\alpha\psi\alphaj+\Gammajk\alpha\psii\alpha+hij\alphac'k\alpha\labeleq:second

(\theequation)

Proposision 3. Given {h_ijk, \psi_ij}, we can calculate {\Gamma_ij^k, b_ij, c'_ij} from , .

Theorem. Given differential forms h, \psi on U satisfying the structure equations, there exists unique (up to isomorphism Fv=Av+a, a in V, A in G₂ⁿ, a, A - const) immersion f:U --> V (defined in small neighborhood of a given point u₀ in U) such that h, \psi defined by , coinside with given one.

Jacobson N. Cayley numbers and normal simple Lie algebgas of type G.- Duke Math.J., 1939, 5, p.775-783.

Zhelvakov K.A., Slin'ko A.M., Shestakov I.P., Shirshov A.I. Kol'za blizkie k assoziativnym. Moskva, Nauka, 1978.

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-82.