Weak holonomy in Dimension 16
by
Thomas Friedrich
Humboldt-Universitaet zu Berlin, Germany

Plenary Lecture

The aim of this lecture is to present a weak holonomy concept associated to the Lie group Spin(9). The spin representation of the group Spin(9) is real and 16-dimensional. According to Berger's holonomy theorem, Spin(9) can occur as the holonomy group of a 16-dimensional Riemannian manifold. However, D. Alekseevski (1968) and R. Brown/ A. Gray (1972) proved that any complete 16-dimensional Riemannian manifold whose holonomy group is contained in Spin(9) is necessarily flat or isometric to the Cayley plane F₄ / Spin(9) or its non compact dual F₄^* / Spin(9).

In 1971 A. Gray introduced the concept of weak holonomy. He proved that if a manifold has one of the groups

G=SO(n),     SU(n),     Sp(n) ·Sp(1),     Sp(n) ·SO(2),     Sp(n),     Spin(7)

as weak holonomy group, then its holonomy is in fact already contained in G. Consequently, only 3 groups may admit a weak holonomy concept which is more general than the traditional holonomy approach:

G = U(n),     G=G2   in dimension 7 ,     G = Spin(9)  in dimension 16.

The first two cases yield a rich geometric structure both as weak and as classical holonomy groups and have intensively been studied. Manifolds with weak holonomy group U(n) are called nearly Kähler. A. Gray investigated them since 1976 and pointed out that they have special properties in dimension 6. This effect is closely related to the fact that on a 6-dimensional manifold, the existence of a nearly Kähler structure is equivalent to the existence of a real Killing spinor (Grunewald 1990). In 1981 S. Marchiafava characterized 7-dimensional manifolds with weak holonomy group G₂, and M. Fernandez / A. Gray studied systematically the different geometric types of G₂-structures. In particular, nearly parallel G₂-structures correspond again to real Killing spinors. Only the case of weak holonomy Spin(9) on 16-dimensional manifolds has been neglected until now.

We will first define a (topological) Spin(9)-structure on a 16-dimensional manifold as some 9-dimensional subbundle V⁹ of the bundle of endomorphisms End (T(M¹⁶)). Locally there exist nine endomorphisms I_\alpha in \Gamma(V⁹)   (1 <= \alpha <= 9) satisfying the relations

I\alpha2 = Id     ,     I\alpha* = I\alpha     ,     I\alpha I\beta = - I\beta I\alpha     for \alpha =/= \beta.

From this point of view a Spin(9)-structure is a 16-dimensional analogon of a quaternionic structure. It was already known that there exists a Spin(9)-invariant and self-dual 8-form \Omega⁸ on R¹⁶. Although it cannot be used to uniquely characterize the structures we are interested in, it will play an important role.

We construct several example of 16-dimensional manifolds admitting natural topological Spin(9)-structures. Then we derive necessary conditions for the Stiefel-Whitney and the Pontrjagin classes of a compact manifold admitting a Spin(9)-reduction of the frame bundle. For example, the complete intersection of three quadrics in P¹¹(C) satisfies all these conditions. The space X⁸⁴ = SO(16)/ Spin(9) and its homotopy type describes in general the obstructions for the existence of a Spin(9)-structure. Using recent results on the homotopy groups \pi_i (SO(n)) outside the stable range we compute \pi_i (X⁸⁴) for i = 1, ... , 14.

We associate to any Spin(9)-reduction a 1-form \Gamma with values in the bundle \Lambda³ (V⁹),

\Gamma in \Lambda1 (M16) \otimes\Lambda3 (V9).

The space \Lambda¹ (R¹⁶) \otimes\Lambda³ (R⁹) decomposes under the action of \sneun into 4 irreducible summands. Depending on the algebraic type of \Gamma there are 16 different geometric types of Spin(9)-structures. One of the components in the splitting of \Lambda¹ (R¹⁶) \otimes\Lambda³ (R⁹) in the representation \Lambda¹ (R¹⁶) itself. This observations yields to the defintion of a particular class of Spin(9)-structures. A topological Spin(9)-structure is called nearly parallel if and only if \Gamma is a vector field. For example, S¹ ×S¹⁵ admits a nearly parallel Spin(9)-structure. Using the fact that the Spin(9)-representation \Lambda⁷ (R¹⁶) is multiplicity-free, we can prove that the 8-form \Omega⁸ of a nearly parallel Spin(9)-structure satisfies the equations

\delta\Omega8 = - 504 (\Gamma\lrcorner \Omega8)     ,    d \Omega8 = - 504 * (\Gamma\lrcorner \Omega8).

The other geometric types of Spin(9)-structures yield similar differential equations.

We sketch the twistor theory for nearly parallel Spin(9)-structures. The space T₁ of all complex structures in \Lambda² (R⁹)

T1 = { J = 9 å \alpha, \beta = 1  x\alpha\beta I\alpha I\beta :     J2 = - Id }

is isomorphic to a complex quadric Q in P⁸ (C). Moreover, the group Spin(9) acts transitively on T₁. Using T₁ as a typical fibre we introduce a twistor space T₁ (M¹⁶) for any 16-dimensional manifold with a fixed Spin(9)-structure. It has a canonical almost complex structure as well as an anti-holomorphic involution without fixed points. From the general theory of twistor spaces we know that the almost complex structure on T₁ (M¹⁶) has to satisfy two integrability conditions. The first one concerns the torsion tensor and turns out to be automatically satisfied in case of a nearly parallel Spin(9)-structure. Therefore, the integrability condition is a relation between the Riemannian curvature \Omega^Z and the derivative D^Z (\Gamma). For example, the twistor space of the Cayley plane F₄ / Spin(9) (\Gamma = 0) is isomorphic to

T1 (F4 / Spin(9))=F4 / (Spin(2) ×Z2 Spin(7)).

Since Spin(2) ×_Z2 Spin(7) is the centralizer of the subgroup Spin(2) in F₄, T₁ (F₄ / Spin(9)) is a generalized flag manifold and therefore a complex projective variety. The twistor space of S¹ ×S¹⁵ with its invariant nearly parallel Spin(9)-structure is a complex subvariety of the twistor space of S¹ ×S¹⁵ considered as a conformally flat Riemannian manifold.

We describe the twistor space of the flat manifold R¹⁶ as an 8-dimensional holomorphic vector bundle N over the quadric Q, and compute its Chern classes as well as the space of all holomorphic sections H⁰ (N) of this bundle. This result allows not only the description of T₁ (R¹⁶), but also of the normal bundle N to any fibre inside an arbitrary twistor space T₁ (M¹⁶).

References:

Thomas Friedrich, Weak Spin(9)-structures on 16-dimensional manifolds, to appear in Äsian Journal of Mathematics".

http://www-irm.mathematik.hu-berlin.de/~friedric

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-71.