de Havilland Mosquito Mk. IV by Michael J. F. Bowyer

By Michael J. F. Bowyer

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11. Die Gegeniiberstellung von Topologie und abstrakter Algebra hat uns zu bemerkenswerten Ergebnissen gefiihrt und eigenartige Beziehungen zwischenden beiden aufgedeckt. Verschiedenen anschaulichen und formalen Bediirfnissen entspringend, erfiillen die beiden Denkweisen verschiedene Funktionen und sind der Ausdruck zweier verschiedener Wege des Verstehens und der 26 Topologie und Algebra Anschauung; sie erganzen und unterstiitzen einander dabei aufs schonste. Aber die Verschiedenheit der beiden ist nur scheinbar; ihre Synthese ergibt nicht nur ein geschlossenes Ganzes, sondern lasst sie gewissermassen ineinander aufgehen.

I do not know of an example of an almost complex manifold which does not admit a complex structure, but probably there are such. 2. Let M be almost complex and v(x) a continuous nonzei^o vector field on M with one possible singularity; then Ixv(x) = w{x) is a second field with one possible singularity at the same point and such that w(x) and v{x) are independent eveiywhere else since ll = — 1 . Therefore there exists on M a 2-field with one isolated singularity, which is obviously of a quite special nature.

2n real, dimensions) have a natural complex structure, as well as Pn itself, and are therefore complex manifolds—^provided that there are no "singularities". There exist, on the other hand, examples of complex manifolds which cannot be imbedded in a Pn (cf. 4). The concept of a complex manifold is therefore more general than that of an algebraic variety. I t is probably also more general than that of a higher-dimensional Riemann surface in a concrete sense, defined as the space of all function elements of an algebroid function on some standard space S\ but such a statement depends of course on the space S and on uniformization possibilities.

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