Differential geometry and topology by Jacob T Schwartz

By Jacob T Schwartz

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2 Generation of the Lie Sphere Group by Inversions πx =x− 2 x, ξ ξ . , those which have lightlike poles. It is clear that π fixes every point in π and that π ξ = −ξ . A direct computation shows that π is in O(n − k, k) and that 2π = I . 6 below concerning the special case of Rk2k , where the metric has signature (k, k). In that case, let {e1 , . . , e2k } be an orthonormal basis with e1 , . . , ek spacelike and ek+1 , . . , e2k timelike. One can naturally choose a basis {v1 , . . , vk , w1 , .

A similar proof shows that Avi , Avj = 0 for i = j . Therefore, the equation Ax, Ay = λ x, y holds on an orthonormal basis, so it holds for all vectors. , λ > 0. 2. In the case k = n − k, conclusion (b) does not necessarily hold. For example, the linear map T defined by T vi = wi , T wi = vi , for 1 ≤ i ≤ k, preserves lightlike vectors, but the corresponding λ = −1. 1 we immediately obtain the following corollary. 3. (a) The group G of Lie sphere transformations is isomorphic to O(n + 1, 2)/{±I }.

When interpreted as a map on the space of spheres, it takes a sphere with center p and signed radius r to the sphere with center µp and signed radius µr. Thus Sµ is one of the two affine Laguerre transformations induced from the Euclidean central dilatation p → µp, for p ∈ R n . The transformation Sµ preserves the sign of the radius and hence the orientation of each sphere in R n . The other affine Laguerre transformation induced from the same central dilatation is Sµ , where is the change of orientation transformation.

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