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Publications [#243395] of Robert Bryant
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 Bryant, RL, Lie groups and twistor spaces,
Duke Math. J., vol. 52 no. 1
(1985),
pp. 223261 [MR87d:58047]
(last updated on 2018/03/18)
Author's Comments: A classification of integrable twistor spaces of various
kinds over
Riemannian symmetric spaces. Given an even
dimensional Riemannian manifold
N, the bundle J(N) over N of orthogonal complex
structures on the tangent
spaces of N has a natural almost complex structure
and complex horizonal
plane field. Unless N has constant sectional curvature,
however, the almost
complex structure on J(N) will not be integrable. A
twistor subspace Z
of J(N) is a subbundle that is an almost complex
submanifold of J(N) and
that has the additional properties that the induced
almost complex structure
on Z is actually integrable and that the horizontal plane
field is tangent
to Z.
In this paper, I find all of the twistor subspaces of
J(N) when N is
a Riemannian symmetric space. These all turn out to
be orbits of the isometry
group of N and so can be classified by examining the
root systems of the
simple Lie groups.
In the last section, I construct a different sort of
twistor space over
each Riemannian symmetric space.
For further developments in this area, consult the
works of F. Burstall,
J. Rawnsley, S. Salamon, and J. Wood.


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