Conjuncted: Philosophizing Twistors via Fibration. Note Quote.


The fibration is not holomorphic (even when this makes sense) but the fibres are complex submanifolds of the twistor space. Let us now see how we might build such fibrations over more general Riemannian manifolds.

So let N be a 2n-dimensional Riemannian manifold. We may at least construct such a fibration of an almost complex manifold over N as follows: let π:J(N) → N be the bundle of almost Hermitian structures of N. Thus the fibre at x ∈ N is

Jx(N) = {j ∈ End(TxN): j2 = −1, j skew-symmetric}.

This bundle is associated to the orthonormal frame bundle of N with typical fibre J(R2n) = O(2n)/U(n) which is a Hermitian symmetric space (in fact it is two disjoint copies of the compact irreducible Hermitian symmetric space SO(2n)/U(n)). In particular, the typical fibre has an O(2n)-invariant complex structure and thus the vertical distribution V = ker dπ inherits an almost complex structure JV. The Levi-Civita connection on the orthonormal frame bundle induces a horizontal distribution H on J(N) so that we have a splitting

T J (N ) = V ⊕ H

with dπ giving an isomorphism between H and π−1TN. This enables us to define a tautological almost complex structure JH on H by

JjH = j
and adding this to JV gives us an almost complex structure J = JV ⊕ JH on J(N). By

construction, the fibres of π are almost complex submanifolds with respect to J.

If we make a conformal change of metric on N, the bundle J(N) remains unchanged although the horizontal distribution H will vary. However, despite this, it can be shown that the almost complex structure J is independent of the choice of metric within a conformal class. Thus our construction may be viewed as one in Conformal Geometry.

Having got our almost complex structure, it is natural to ask whether or not it is integrable so that J(N) is an honest complex manifold. For this, of course, it is necessary and sufficient that the Nijenhuis tensor NJ of J vanish. The obstruction to this vanishing lies in the curvature tensor of N [22] :

Theorem: Let j ∈ J(N) with √−1-eigenspace T+ ⊂ Tπ(j)NC. Let R denote the Riemann curvature tensor of N. Then NJ vanishes at j if and only if

R(T+, T+)T+ ⊂ T+

Thus J is integrable if the above equation holds for all maximally isotropic subspaces T+ of TNC. This is a condition on the curvature tensor that can be analysed in terms of the representation theory of O(2n) on the space of curvature tensors and one concludes:

Corollary: J is integrable if and only if the Weyl tensor of R vanishes identically (i.e. N is locally conformally flat).

Thus J(N) is a complex manifold only in extremely restricted circumstances. The moral to be drawn from this is that J(N) is “too big” in general for J to be integrable. It is therefore appropriate to seek subbundles of J(N) picked out by the geometry of N in the hope that some of these are complex manifolds. One way to do this is is to restrict attention to those elements of J(N) that are compatible with the holonomy of N.



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