3. — Complex coordinates in E4.

There is one immediate objection to such a method, namely that it is too closely tied to the particular coordinate system and therefore is unlikely

The introduction of the Penrose space P3(C) as a tool to study problems on $4 (or -R4) can be motivated in the following way. It is very well known that many classical problems in 2 real variables (x, y) are best solved by introducing the single complex variable z = x iy, and then using the powerful methods of holomorphic function theory. When we pass from B2 to jK4 one might naively try a similar device by introducing two complex variables


to give significant results. For example a permutation of the four coordinates xx, x2, x3, xi would lead to new complex variables not well related to the first choice. In B2 on the other hand provided we have fixed the metric and orientation the complex structure is unambiguous. The complex number i is given by rotation through tt/2 in the positive sense.

Since there is no natural choice of complex structure on Bl we can try to consider simultaneously all choices which are compatible with the metric and orientation. Effectively we have to define i as a proper orthogonal trans­formation with i2 = — 1. If we have made one such choice then transform­ing (conjugating) by elements of $0(4) will produce all other choices. More­over transforming by elements of U(2) leaves the first choice unaltered. Hence the set of all complex structures is naturally parametrized by the coset space



Thus to each u e $2 we have a complex structure on Bl, namely an iso­morphism

If we now want to introduce complex variable methods in Rl we need 3 complex variables {u,z",z%): the first variable u tells us which complex structure to use and the next two are the complex coordinates themselves. Since the coordinates zx, z2 depend on u the situation is a little delicate. The Penrose picture clarifies this geometrically as we shall now explain.

For simplicity we shall now work only over Bl c $4 and so we remove the «point at oo» in S1 and correspondingly we remove the projective «line at oo » in P3 that lies over it in the fibration (1.1). Then we get a fibration ■iMO) — Pi{G) Bl. Projective planes in P3(G) which meet in the «line at oo » become «parallel» affine planes in P3(G) — Pi{G). Thus we havo the following picture for our fibration:



Over a given point (say the origin) of Bl we have the fibre Pi(C') = S2 par­ametrized by u. The plane of our parallel system through u is a copy of C2 and under the projection gets identified with B4. This is the way in which Bl gets the complex structure corresponding to u. The fact that this complex structure is changing with u means that the vertical identification between the different G\ does not preserve the complex structure. This can be ex­pressed differently by noting that the «horizontal» projection PS{G)-P1(G)-^- -*-Pi(G) mapping G\ into u is that of a complex vector bundle which is not isomorphic to the product Pi(G) x G2, although the underlying real vector bundle is isomorphic to P1(C) x B4. This can happen because bundles over 82 = Pi(O) are topologically classified by maps of the equa­torial S1 into the group of the bundle and the fundamental groups nx in our case are: and our bundle corresponds to the integer 2 which gives zero in ^($0(4)).

This topological fact therefore lies behind the «linkage» between the complex coordinates z\ and the complex parameter u. Working however in the space P3(0) — Pi(G) we can locally introduce three independent («unlinked») complex variables and use these instead of (u,    Thus

the Penrose space enables us to introduce 3 complex variables in a natural way to study problems in Bl.

The same picture applies infinitesimally for the complete fibration P3(<7) -> Sl. At each point u e P3(0) we can consider the tangent space Tu and the subspace L„ of tangents to the fibre through u. The quotient space TJLU is then a complex vector space which by projection gets naturally identified with the real tangent space to 84 at the point below u.

Note in particular that, since complex manifolds have a natural orienta­tion, inherits a natural orientation from its description as the base of the fibration P3(<7) $4. This is the orientation which we fix (the opposite convention is adopted in [3]).