20 - Semianalytic category

Summary

In this chapter we define semianalytic structures and draw conclusions from those which are important for the uniqueness of the kinematical representation of LQG. Semianalytic structures are intuitively the same thing as piecewise analytic structures, that is, objects such as paths or surfaces are analytic on generic subsets but analyticity may be violated on lower-dimensional subsets. On those subsets there is again a notion of semianalyticity. This enables one to take advantage of analyticity while making the constructions local: for instance, strictly analytical paths are determined everywhere on their analytic extension once they are known on an open set, thus making them very non-local. If we make it semianalytic then these data only determine the path up to the next point where analyticity is reduced to Cⁿ, n > 0. This is important because we need to make sure that certain local constructions do not have an impact on regions far away from the region of interest. We will see this explicitly in the uniqueness proof.

We will now develop elements of semianalytic differential geometry in analogy to Chapter 19. We begin with ℝⁿ with its canonical analytic structure. For general manifolds M we will assume that they are differential manifolds with given smooth structure and that a compatible analytic structure has been fixed. An introduction to semianalytical notions can be found in [888].