The Element of Volume of the Rotation Group

The pseudo-angle characterizes the pseudo-conformal group G." Let S2m be a fixed 2m dimensional manifold contained in 2;2 so that m . n. LetP be an arbitrary point of S2m. Suppose that S2,1n is an arbitrary (2n 1) dimensional manifold in Z2., which contains P, and let the intersection of S2m and S2,-1 be an S2m-j. Thus S2m is not contained in S2,1. The pseudo-angle between any curve C, in S2m, and S2,.1 at P, is equal to that between C and S2m1, for every S2,.1, if and only if S2m is pseudo-conformal. This gives a geometric characterization of the pseudo-conformal manifolds Z2m contained in a given pseudo-conformal manifold Z2, where m _ n. If w = 4 + i4, is monogenic over R, then the curves of k = const. are pseudo-orthogonal to the manifolds P = const. For n = 1, this reduces to the well-known result that the components of a monogenic function of a complex variable give rise to an orthogonal isothermal net.