By Taubes C.H.
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Extra resources for An introduction to bundles, connections, metrics and curvature
Introducing a Minkowski scalar product on the space of homogeneous coordinates the incidence of a point and a sphere, and the intersection angle of hyperspheres can be described conveniently. 2. Hypersphere pencils and hypersphere complexes are introduced as conﬁgurations of hyperspheres that are linear in our projective model space, and their geometry is discussed. These conﬁgurations will, on one hand, become important in our description of the M¨ obius group and, on the other hand, for the description of spheres of arbitrary dimension.
1 Lemma. Any Lorentz transformation µ ∈ O1 (n + 2) descends to a projective transformation 18) µ : P n+1 → P n+1 that preserves S n (as the absolute quadric). Any projective transformation on P n+1 that preserves S n gives rise to n+1 n+1 → PO a M¨ obius transformation µ : S n → S n , with its action µ : PO on the outer space being exactly the action of the M¨ obius transformation on the hyperspheres of S n . 2 Proof. As a linear transformation of n+2 , a Lorentz transformation maps lines in P n+1 to lines, and it preserves S n ⊂ P n+1 as it maps the onto itself.
In this section the announced proof of Liouville’s theorem is given by making use of the description of hyperbolic geometry as a subgeometry of M¨obius geometry. The (global) conformal diﬀeomorphisms of Euclidean and hyperbolic space are discussed. 6. The notions of sphere congruence and of an envelope of a sphere congruence, which are central to many constructions in M¨ obius diﬀerential geometry, are introduced and a convenient formulation for the enveloping condition is given. Conditions on a (hyper)sphere congruence to have one or two envelopes are derived, and the notion of a “strip” is deﬁned.