Submanifolds and special structures on the octonians,
J. Differential Geom., vol. 17 no. 2
(1982),
pp. 185–232 [MR84h:53091]
(last updated on 2004/04/06)
Abstract:
A study of the geometry of submanifolds of real 8-space
under the
group of motions generated by translations and
rotations in the subgroup
Spin(7) instead of the full SO(8). I call real 8-space
endowed with this
group
O or octonian space.
The fact that the stabilizer of an oriented 2-plane in
Spin(7) is U(3)
implies that any oriented 6-manifold in O
inherits a U(3)-structure.
The first part of the paper studies the generality of the
6-manifolds whose
inherited U(3)-structure is symplectic, complex, or
Kähler, etc. by
applying the theory of exterior differential systems.
I then turn to the study of the standard 6-sphere in
O as an
almost complex manifold and study the space of what
are now called pseudo-holomorphic
curves in the 6-sphere. I prove that every compact
Riemann surface occurs
as a (possibly ramified) pseudo-holomorphic curve in
the 6-sphere. I also
show that all of the genus zero pseudo-holomorphic
curves in the 6-sphere
are algebraic as surfaces.
Reprints are available.
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