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By Michael Spivak

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Additional resources for A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition

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Suppose that [x] is timelike. Then the Lie metric , has signature (n + 1, 1) on the vector space x ⊥ , and x ⊥ contains the two-dimensional lightlike vector space that projects to . 3. The next result establishes the relationship between the points on a line in Qn+1 and the corresponding parabolic pencil of spheres in R n . 6. , k1 , k2 = 0. (b) If the line [k1 , k2 ] lies on Qn+1 , then the parabolic pencil of spheres in R n corresponding to points on [k1 , k2 ] is precisely the set of all spheres in oriented contact with both [k1 ] and [k2 ].

In that case, we let I1 be the identity transformation. Next, the parallel transformation P−r takes the point [y] to the point [z] with zn+3 = 0 corresponding to the point sphere p in R n . Finally, an inversion I2 in a 48 3 Lie Sphere Transformations sphere centered at p takes [z] to the improper point [e1 −e2 ]. Since the transformation I2 P−r I1 α takes [e1 − e2 ] to itself, it is a Laguerre transformation ψ. Since each inversion is its own inverse and the inverse of P−r is Pr , we have α = I1 Pr I2 ψ, a product of Laguerre and Möbius transformations.

Note that Pt is a Laguerre transformation that is not a Möbius transformation. It takes point spheres to spheres with signed radius t and takes spheres of radius −t to point spheres. Thus, the group of Laguerre translations is a commutative subgroup of the group of affine Laguerre transformations, and we have shown that it decomposes as follows. 15. Any Laguerre translation T can be written in the form T = Pt τv , where Pt is a parallel transformation, and τv is the orientation preserving Laguerre translation induced by Euclidean translation by the vector v.

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