Math @ Duke

Publications [#243390] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, Submanifolds and special structures on the octonians,
Journal of Differential Geometry, vol. 17 no. 2
(January, 1982),
pp. 185232, International Press of Boston [MR84h:53091], [doi]
(last updated on 2019/05/22)
Abstract: A study of the geometry of submanifolds of real 8space
under the
group of motions generated by translations and
rotations in the subgroup
Spin(7) instead of the full SO(8). I call real 8space
endowed with this
group
O or octonian space.
The fact that the stabilizer of an oriented 2plane in
Spin(7) is U(3)
implies that any oriented 6manifold in O
inherits a U(3)structure.
The first part of the paper studies the generality of the
6manifolds 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 6sphere in
O as an
almost complex manifold and study the space of what
are now called pseudoholomorphic
curves in the 6sphere. I prove that every compact
Riemann surface occurs
as a (possibly ramified) pseudoholomorphic curve in
the 6sphere. I also
show that all of the genus zero pseudoholomorphic
curves in the 6sphere
are algebraic as surfaces.
Reprints are available.


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