By W. R. Knorr
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Additional info for The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry
Example text
Namely, if we divide G by the maximal compact subgroup flcfi then, by the compactness of H, the spact V = G/H admits a G-invariant metric g. Such a metric is complete (this is elementary) and by a well known theorem of E. Cartan K(g) < 0. This curvature is strictly negative if and only if r a n k ^ G — 1 , and K(g) is constant if and only if G is locally isomorphic to O(n, 1). The compact manifolds V covered by V are associated to discrete subgroups re G which are usually produced by arithmetic constructions.
The sectional curvature K(v) of V c: 1ft3 equals the Jaeobian of the Gauss map V —> Ss at v, or equivalently to the •product of the principal curvatures {eienvahies of TIy or of the shape operator A) at v. Of course, the proof is trivial by the standards of the modern infinitesimal caulculus. Yet, the major consequence of the theorem looks as remarkable as it appeared 200 years ago: the Jaeobian of the Gauss map does not change if we bend V in IR^, that is if we apply a deformation preserving the length of the curves in V.
Furthermore, if dc(W) develops a self-inter section without focal points, then V~ becomes locally represented as an intersection of smooth convex subsets and so again it is convex. Then it is easy to believe in the convexity at the focal points aa well as these are just « infinitesimal » double points (vanishing of the differential of a map at a tangent vector i 6 T(W) brings together the « infinitely closed points » corresponding to the « two ends » of T). SIGN AND GEOMETRIC MEANING OF CURVATURE 53 To make the above rigorous, one may use a piecewise smooth approximation (compare § 0) of convex hypersurfaces (and subsets) as in Fig.